A Quadratic Programming Bibliography

A method for restoring an optical image which is subjected to low-pass frequency filtering is presented. It is assumed that the object whose image is restored is of finite spatial extent. The problem is treated as an algebraic image-restoration problem which is then solved as a quadratic programming problem with bounded variables. The regularization technique for the ill-posed system is to replace the consistent system of the quadratic programming problem by an approximate system of smaller rank. The rank which gives a best or near-best solution is estimated. This method is a novel one, and it compares favorably with other known methods. Computer-simulated examples are presented. Comments and conclusions are given. N. N. Abdelmalek and T. Kasvand. Digital image restoration using quadratic programming. Applied Optics, 19(19), 3407–3415, 1980. Abstract. The problem of digital image restoration is considered by obtaining an approximate solution to the Fredholm integral equation of the first kind in two variables. The system of linear equations resulting from the discretization of the integral equation is converted to a consistent system of linear equations. The problem is then solved as a quadratic programming problem with bounded variables where the unknown solution is minimized in the L2 norm. In this method minimum computer storage is needed, and the repeated solutions are obtained in an efficient way. Also the rank of the consistent system which gives a best or near best solution is estimated. Computer simulated examples using spatially separable pointspread functions are presented. Comments and conclusions are given. The problem of digital image restoration is considered by obtaining an approximate solution to the Fredholm integral equation of the first kind in two variables. The system of linear equations resulting from the discretization of the integral equation is converted to a consistent system of linear equations. The problem is then solved as a quadratic programming problem with bounded variables where the unknown solution is minimized in the L2 norm. In this method minimum computer storage is needed, and the repeated solutions are obtained in an efficient way. Also the rank of the consistent system which gives a best or near best solution is estimated. Computer simulated examples using spatially separable pointspread functions are presented. Comments and conclusions are given. R. A. Abrams and A. Ben Israel. A duality theorem for complex quadratic programming. Journal of Optimization Theory and Applications, 4(4), 245–252, 1969. Abstract. A duality theory for complex quadratic programming over polyhedral cones is developed, following Dorn, by using linear duality theory. A duality theory for complex quadratic programming over polyhedral cones is developed, following Dorn, by using linear duality theory. J. W. Adams. Quadratic programming approaches to new optimal windows and antenna arrays. Conference Record. Twenty Fourth Asilomar Conference on Signals, Systems and Computers Maple Press, San Jose, CA, USA, 1, 69–72, 1990a. Abstract. New window design problems are formulated in terms of quadratic programming. The new windows permit the designer to control the tradeoff between the peak sidelobe level and the total sidelobe energy. In addition, linear constraints can be imposed on the design problem. The proposed methods are applicable to applications in the fields of signal processing and antenna arrays. New window design problems are formulated in terms of quadratic programming. The new windows permit the designer to control the tradeoff between the peak sidelobe level and the total sidelobe energy. In addition, linear constraints can be imposed on the design problem. The proposed methods are applicable to applications in the fields of signal processing and antenna arrays. J. W. Adams. Quadratic programming approaches to new problems in digital filter design. Conference Record. Twenty Fourth Asilomar Conference on Signals, Systems and Computers Maple Press, San Jose, CA, USA, 1, 307–310, 1990b. Abstract. New digital filter design problems are formulated in terms of quadratic programming. The new filters permit the designer to control the tradeoff between the peak errors and the total squared errors. In New digital filter design problems are formulated in terms of quadratic programming. The new filters permit the designer to control the tradeoff between the peak errors and the total squared errors. In 2 A QUADRATIC PROGRAMMING BIBLIOGRAPHY particular, the mean-squared error can be minimized subject to peak errors constraints, as required in many practical design problems. In addition, equality constraints can be imposed for special applications. J. W. Adams, P. Kruethong, R. Hashemi, J. L. Sullivan, and D. R. Gleeson. New quadratic programming algorithms for designing FIR digital filters. Conference Record of The Twenty Seventh Asilomar Conference on Signals, Systems and Computers. IEEE Comput. Soc Press, Los Alamitos, CA, USA, 2, 1206–1210, 1993. Abstract. The Parks-McClellan algorithm (1973) is very popular for designing FIR digital filters. It is based on a linear programming algorithm called the Remez exchange. Our new algorithm is based on quadratic programming, which includes linear programming as a special case. The filters in this paper permit the designer to control the tradeoff between the peak error and the total squared error. These filters are designed according to the peak-constrained least-squares (PCLS) optimality criterion. The Parks-McClellan algorithm (1973) is very popular for designing FIR digital filters. It is based on a linear programming algorithm called the Remez exchange. Our new algorithm is based on quadratic programming, which includes linear programming as a special case. The filters in this paper permit the designer to control the tradeoff between the peak error and the total squared error. These filters are designed according to the peak-constrained least-squares (PCLS) optimality criterion. J. W. Adams, J. L. Sullivan, D. R. Gleeson, P. H. Chang, and R. Hashemi. Application of quadratic programming to FIR digital filter design problems. Conference Record of the Twenty Eighth Asilomar Conference on Signals, Systems and Computers. IEEE Comput. Soc Press, Los Alamitos, CA, USA, 1, 314–318, 1994. Abstract. Quadratic programming problems have long been of interest in the business community. Quadratic programming is often used as the basis for ”program trading” where stocks are automatically bought and sold by mutual funds to optimize profits. Quadratic programming algorithms can also be used to optimize digital filters, as discussed in this paper. We present the generalized multiple exchange (GME), simplified generalized multiple exchange (SGME) and modified generalized multiple exchange (MGME) algorithms for designing constrained least-squares (CLS) filters. The CLS filters are generalizations of the popular minimax and least-squares filters. The CLS filters are important not only because of their generality, but also because they are needed for many practical applications. Quadratic programming problems have long been of interest in the business community. Quadratic programming is often used as the basis for ”program trading” where stocks are automatically bought and sold by mutual funds to optimize profits. Quadratic programming algorithms can also be used to optimize digital filters, as discussed in this paper. We present the generalized multiple exchange (GME), simplified generalized multiple exchange (SGME) and modified generalized multiple exchange (MGME) algorithms for designing constrained least-squares (CLS) filters. The CLS filters are generalizations of the popular minimax and least-squares filters. The CLS filters are important not only because of their generality, but also because they are needed for many practical applications. W. P. Adams and P. M. Dearing. On the equivalence between roof duality and Lagrangian duality for unconstrained 0–1 quadratic programming problems. Discrete Applied Mathematics, 48(1), 1–20, 1994. Abstract. Considers techniques for computing upper bounds on the optimal objective function value to any unconstrained 0–1 quadratic programming problem (maximization). In particular, the authors study three methods for obtaining upper bounds as presented in a recent paper by Hammer, Hansen, and Simeone (1984) (1) generating two classes of upper-bounding linear functions referred to as paved upper planes and roofs, (2) solving the continuous relaxation of a mixed-integer linear problem by Rhys (1970), and (3) the quadratic complementation of variables which results in a bound called the height. The authors show that all three methods directly result from standard properties of a reformulation of the quadratic problem as a mixedinteger linear program, with methods (1) and (3) resulting from a Lagrangian dual of this reformulation. Based on this reformulation, they expand upon the published results. Considers techniques for computing upper bounds on the optimal objective function value to any unconstrained 0–1 quadratic programming problem (maximization). In particular, the authors study three methods for obtaining upper bounds as presented in a recent paper by Hammer, Hansen, and Simeone (1984) (1) generating two classes of upper-bounding linear functions referred to as paved upper planes and roofs, (2) solving the continuous relaxation of a mixed-integer linear problem by Rhys (1970), and (3) the quadratic complementation of variables which results in a bound called the height. The authors show that all three methods directly result from standard properties of a reformulation of the quadratic problem as a mixedinteger linear program, with methods (1) and (3) resulting from a Lagrangian dual of this reformulation. Based on this reformulation, they expand upon the published results. W. P. Adams and H. D. Sherali. A tight linearization and an algorithm for 0–1 quadratic programming problems. Management Science, 32(10), 1274–1290, 1986. Abstract. The paper is concerned with the solution of linearly constrained 0–1 quadratic programming problems. Problems of this kind arise in numerous economic, location decision, and strategic planning situations, including capital budgeting, facility location, quadratic assignment, media selection, and dynamic set covering. A new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods. An implicit enumeration algorithm which uses Lagrangian relaxation, Benders’ cutting planes, and local explorations is designed to exploit the strength of this linearization. Computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm. The paper is concerned with the solution of linearly constrained 0–1 quadratic programming problems. Problems of this kind arise in numerous economic, location decision, and strategic planning situations, including capital budgeting, facility location, quadratic assignment, media selection, and dynamic set covering. A new linearization technique is presented for this problem which is demonstrated to yield a tighter continuous or linear programming relaxation than is available through other methods. An implicit enumeration algorithm which uses Lagrangian relaxation, Benders’ cutting planes, and local explorations is designed to exploit the strength of this linearization. Computational experience is provided to demonstrate the usefulness of the proposed linearization and algorithm. S. N. Afriat. The quadratic form definite on a manifold. Proceedings of the Cambridge Philosophical Society, 47, 1–6, 1951. N. I. M. GOULD & PH. L. TOINT 3 A. Aggarwal and C. A. Floudas. A decomposition approach for global optimum search in QP, NLP and MINLP problems. Annals of Operations Research, 25(1–4), 119–145, 1990. Abstract. A new approach for global optimum search is presented which involves a decomposition of the variable set into two sets-complicating and noncomplicating variables. This results in a decomposition of the constraint set leading to two subproblems. The decomposition of the original problem induces special structure in the resulting subproblems and a series of these subproblems are then solved, using the generalised Benders’ decomposition technique, to determine the optimal solution. Mathematical properties of the proposed approach are presented. Even though the proposed approach cannot guarantee the determination of the global optimum, computational experience on a number of nonconvex QP, NLP and MINLP example problems indicates that a global optimum solution can be obtained from various starting points. A new approach for global optimum search is presented which involves a decomposition of the variable set into two sets-complicating and noncomplicating variables. This results in a decomposition of the constraint set leading to two subproblems. The decomposition of the original problem induces special structure in the resulting subproblems and a series of these subproblems are then solved, using the generalised Benders’ decomposition technique, to determine the optimal solution. Mathematical properties of the proposed approach are presented. Even though the proposed approach cannot guarantee the determination of the global optimum, computational experience on a number of nonconvex QP, NLP and MINLP example problems indicates that a global optimum solution can be obtained from various starting points. F. G. Akhmadov. Computational method for solving the quadratic programming problem in L2 0 T space. Izvestiya Akademii Nauk Azerbaidzhanskoi SSR, Seriya Fiziko– Tekhnicheskikh i Matematicheskikh Nauk, 4(4), 102–106, 1983. Abstract. A method for solving the quadratic programming problem in the space of all vector-functions, each component square of which is integrable, is given. It is supposed that the matrix of the quadratic form is positively defined. A method for solving the quadratic programming problem in the space of all vector-functions, each component square of which is integrable, is given. It is supposed that the matrix of the quadratic form is positively defined. F. A. Al-Khayyal. Linear, quadratic and bilinear programming approaches to the linear complementarity problems. European Journal of Operations Research, 24, 216–227, 1987. F. A. Al-Khayyal. Jointly constrained bilinear programs and related problems: An overview. Computers in Mathematical Applications, 19, 53–62, 1990. F. A. Al-Khayyal and J. E. Falk. Jointly constrained biconvex programming. Mathematics of Operations Research, 8, 273–286, 1983. C. Alessandri and A. Tralli. Frictionless contact with BEM using quadratic programming— discussion. Journal of Engineering Mechanics-ASCE, 119(12), 2538–2540, 1993. B. Alidaee, G. A. Kochenberger, and A. Ahmadian. 0–1 quadratic programming approach for optimum solutions of two scheduling problems. International Journal of Systems Science, 25(2), 401–408, 1994. Abstract. Two scheduling problems are considered: (1) scheduling n jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET); (2) scheduling n jobs non-preemptively on two parallel identical processors to minimize weighted mean flow time. In the second problem, a pre-ordering of the jobs is assumed that must be satisfied for any set of jobs scheduled on each specific machine. Both problems are known to be NP-complete. A 0–1 quadratic assignment formulation of the problems is presented. An equivalent 0–1 mixed integer linear programming approach for the problems are considered and a numerical example is given. The formulations presented enable one to use optimal and heuristic available algorithms of 0–1 quadratic assignment for the problems considered here. Two scheduling problems are considered: (1) scheduling n jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET); (2) scheduling n jobs non-preemptively on two parallel identical processors to minimize weighted mean flow time. In the second problem, a pre-ordering of the jobs is assumed that must be satisfied for any set of jobs scheduled on each specific machine. Both problems are known to be NP-complete. A 0–1 quadratic assignment formulation of the problems is presented. An equivalent 0–1 mixed integer linear programming approach for the problems are considered and a numerical example is given. The formulations presented enable one to use optimal and heuristic available algorithms of 0–1 quadratic assignment for the problems considered here. K. S. Alsultan and K. G. Murty. Exterior point algorithms for nearest points and convex quadratic programs. Mathematical Programming, 57(2), 145–161, 1992. A. Altman. QHOPDM—a higher-order primal-dual method for large-scale convex quadratic programming. European Journal of Operational Research, 87(1), 200–202, 1995. A. Altman. Higher order primal-dual interior point method for separable convex quadratic optimization. Control and Cybernetics, 25(4), 761–772, 1996. 4 A QUADRATIC PROGRAMMING BIBLIOGRAPHY A. Altman and J. Gondzio. Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Logilab Technical Report 1998.6, Department of Management Sciences, University of Geneva, Geneva, Switzerland, 1998. Abstract. This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidefinite systems known to be strongly factorizable to a form of Cholesky-like factorization. Two different regularization techniques, primal and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs. This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidefinite systems known to be strongly factorizable to a form of Cholesky-like factorization. Two different regularization techniques, primal and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs. H. Amato and G. Mensch. Rank restrictions on the quadratic form in indefinite quadratic programming. Unternehmensforschung, 15(3), 214–216, 1971. Abstract. A quadratic programming problem, where q x aT x xT Qx is an indefinite objective function, can be solved with Swarup’s approach to optimizing cT x α dT x β only if the rank of Q is two; if Q is definite, the rank of Q must be one. D. E. Amos and M. L. Slater. Polynomial and spline approximation by quadratic programming. Communications of the ACM, 12(7), 379–381, 1969. See also, Collected Algorithms from ACM, 1985. Abstract. The problem of approximation to a given function, or of fitting a given set of data, where the approximating function is required to have certain of its derivatives of specified sign over the whole range of approximation, is studied. Two approaches are presented, in each of which quadratic programming is used to provide both the constraints on the derivatives and the selection of the function which yields the best fit. The first is a modified Bernstein polynomial scheme, and the second is a spline fit. The problem of approximation to a given function, or of fitting a given set of data, where the approximating function is required to have certain of its derivatives of specified sign over the whole range of approximation, is studied. Two approaches are presented, in each of which quadratic programming is used to provide both the constraints on the derivatives and the selection of the function which yields the best fit. The first is a modified Bernstein polynomial scheme, and the second is a spline fit. P. Anand. Decomposition principle for indefinite quadratic programe. Trabajos de Estadistica y de Investigacion, 23, 61–71, 1972. S. C. Anand, F. E. Weisgerber, and H. S. Hwei. Direct solution vs quadratic programming technique in elastic-plastic finite element analysis. Computers and Structures, 7(2), 221– 228, 1977. Abstract. Elastic-plastic plane stress finite element analysis of a disk rolling on a rigid track is performed by the direct method as well as the quadratic programming technique. Tresca and von Mises’ yield conditions are used in the former whereas an approximate piecewise linear Tresca yield condition is used in the later case. It is concluded from a comparison of the computer times needed in the two cases that the direct method is far superior to the quadratic programming technique. Elastic-plastic plane stress finite element analysis of a disk rolling on a rigid track is performed by the direct method as well as the quadratic programming technique. Tresca and von Mises’ yield conditions are used in the former whereas an approximate piecewise linear Tresca yield condition is used in the later case. It is concluded from a comparison of the computer times needed in the two cases that the direct method is far superior to the quadratic programming technique. A. Anckonie. A quadratic program for determining efficient frontier portfolio compositions using the SAS language. In ‘SUGI 10—Proceedings of the Tenth Annual SAS Users Group International Conference’, Vol. 15, pp. 55–60, 1985. W. Anfiloff. Gravity interpretation with the aid of quadratic programming. Geophysics, 46(3), 340–341, 1981. P. L. De Angelis, P. M. Pardalos, and G. Toraldo. Quadratic programming with box constraints. In I. M. Bomze, ed., ‘Developments in Global optimization’, pp. 73–93, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. N. I. M. GOULD & PH. L. TOINT 5 K. M. Anstreicher. On long step path following and SUMT for linear and quadratic programming. SIAM Journal on Optimization, 6(1), 33–46, 1996. Abstract. We consider a long step barrier algorithm for the minimization of a convex quadratic objective subject to linear inequality constraints. The algorithm is a dual version of a method developed by K. M. Anstreicher et al. (1993), and requires O nL or O sqrtnL iterations to solve a problem with n constraints, depending on how the barrier parameter is reduced. As a corollary of our analysis we demonstrate that the classical SUMT algorithm, exactly as implemented in 1968, solves linear and quadratic programs in O nLlogL iterations, with proper initialization and choice of parameters. K. M. Anstreicher. On the equivalence of convex programming bounds for boolean quadratic programming. Working paper, Department of Management Science, University of Iowa, Iowa City, USA, 1997. We consider a long step barrier algorithm for the minimization of a convex quadratic objective subject to linear inequality constraints. The algorithm is a dual version of a method developed by K. M. Anstreicher et al. (1993), and requires O nL or O sqrtnL iterations to solve a problem with n constraints, depending on how the barrier parameter is reduced. As a corollary of our analysis we demonstrate that the classical SUMT algorithm, exactly as implemented in 1968, solves linear and quadratic programs in O nLlogL iterations, with proper initialization and choice of parameters. K. M. Anstreicher. On the equivalence of convex programming bounds for boolean quadratic programming. Working paper, Department of Management Science, University of Iowa, Iowa City, USA, 1997. Abstract. Recent papers have shown the equivalence of several tractable bounds for Boolean quadratic programming. In this note we give simplified proofs for these results, and also show that all of the bounds considered are simultaneously attained by one diagonal perturbation of the quadratic form. K. M. Anstreicher and N. W. Brixius. Solving quadratic assignment problems using convex quadratic programming relaxations. Working paper, Department of Management Science, University of Iowa, Iowa City, USA, 2000. Recent papers have shown the equivalence of several tractable bounds for Boolean quadratic programming. In this note we give simplified proofs for these results, and also show that all of the bounds considered are simultaneously attained by one diagonal perturbation of the quadratic form. K. M. Anstreicher and N. W. Brixius. Solving quadratic assignment problems using convex quadratic programming relaxations. Working paper, Department of Management Science, University of Iowa, Iowa City, USA, 2000. Abstract. We describe a branch-and-bound algorithm for the quadratic assignment problem (QAP) that uses a convex quadratic programming (QP) relaxation to obtain a bound at each node. The QP subproblems are approximately solved using the Frank-Wolfe algorithm, which in this case requires the solution of a linear assignment problem on each iteration. Our branching strategy makes extensive use of dual information associated with the QP subproblems. We obtain state-of-the-art computational results on large benchmark QAPs. K. M. Anstreicher and N. W. Brixius. A new bound for the quadratic assignment problem based on convex quadratic programming. Mathematical Programming, 89(3), 341–357, 2001. We describe a branch-and-bound algorithm for the quadratic assignment problem (QAP) that uses a convex quadratic programming (QP) relaxation to obtain a bound at each node. The QP subproblems are approximately solved using the Frank-Wolfe algorithm, which in this case requires the solution of a linear assignment problem on each iteration. Our branching strategy makes extensive use of dual information associated with the QP subproblems. We obtain state-of-the-art computational results on large benchmark QAPs. K. M. Anstreicher and N. W. Brixius. A new bound for the quadratic assignment problem based on convex quadratic programming. Mathematical Programming, 89(3), 341–357, 2001. Abstract. We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort. K. M. Anstreicher, D. den Hertog, C. Roos, and T. Terlaky. A long-step barrier method for convex quadratic programming. Algorithmica, 10(5), 365–382, 1993. We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort. K. M. Anstreicher, D. den Hertog, C. Roos, and T. Terlaky. A long-step barrier method for convex quadratic programming. Algorithmica, 10(5), 365–382, 1993. Abstract. In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations is O nL or O nL , depending on how the barrier parameter is updated. K. Aoki and T. Fujikawa. VAR planning and nonconvex quadratic programming. Transactions of the Institute of Electrical Engineers of Japan, 100(3), 78–88, 1980. In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations is O nL or O nL , depending on how the barrier parameter is updated. K. Aoki and T. Fujikawa. VAR planning and nonconvex quadratic programming. Transactions of the Institute of Electrical Engineers of Japan, 100(3), 78–88, 1980. Abstract. So far, many methods based on linear programming, nonlinear programming and integer programming have been proposed for var planning. In the linear programming method, the relationship between voltage, active power and reactive power is represented by linear equations. The nonlinear programming method, in which this relationship is expressed exactly, is not suited for a large-scale power system. Both these methods neglect discrete variables representing the number of the condenser or reactor units and the transformer tap positions. The integer programming method, which enables handling these discrete variables, is of course not suited for a large-scale power system. The authors show that the condenser planning problem is formulated into a parametric convex quadratic programming and the reactor planning problem is formulated into a parametric nonconvex quadratic programming. So far, many methods based on linear programming, nonlinear programming and integer programming have been proposed for var planning. In the linear programming method, the relationship between voltage, active power and reactive power is represented by linear equations. The nonlinear programming method, in which this relationship is expressed exactly, is not suited for a large-scale power system. Both these methods neglect discrete variables representing the number of the condenser or reactor units and the transformer tap positions. The integer programming method, which enables handling these discrete variables, is of course not suited for a large-scale power system. The authors show that the condenser planning problem is formulated into a parametric convex quadratic programming and the reactor planning problem is formulated into a parametric nonconvex quadratic programming. 6 A QUADRATIC PROGRAMMING BIBLIOGRAPHY K. Aoki and M. Kanezashi. A decomposition algorithm for a dual angular type quadratic programming. In A. Lew, ed., ‘Proceedings of the 6th Hawaii International Conference on Systems Sciences. Western Periodicals, North Hollywood, CA, USA’, pp. 358–360, 1973. Abstract. This paper deals with a decomposition procedure for the problem whose objective function is linear for a coupling variable. Moreover it is described by taking advantage of such characteristics that one can easily obtain the variation of a coupling variable by computing linear forms. This paper deals with a decomposition procedure for the problem whose objective function is linear for a coupling variable. Moreover it is described by taking advantage of such characteristics that one can easily obtain the variation of a coupling variable by computing linear forms. K. Aoki and T. Satoh. Economic dispatch with network security constraints using parametric quadratic programming. IEEE Transactions on Power Apparatus and Systems, PAS101(12), 4548–4556, 1982. Abstract. This paper presents an efficient method to solve an economic load dispatch problem with DC load flow type network security constraints. The conventional linear programming and quadratic programming methods cannot deal with transmission losses as a quadratic form of generator outputs. In order to overcome this defect, the extension of the quadratic programming method is proposed, which is designated as the parametric quadratic programming method. The upper bounding technique and the relaxation method are coupled with the proposed method for the purpose of computational efficiency. The test results show that the proposed method is practical for real-time applications. This paper presents an efficient method to solve an economic load dispatch problem with DC load flow type network security constraints. The conventional linear programming and quadratic programming methods cannot deal with transmission losses as a quadratic form of generator outputs. In order to overcome this defect, the extension of the quadratic programming method is proposed, which is designated as the parametric quadratic programming method. The upper bounding technique and the relaxation method are coupled with the proposed method for the purpose of computational efficiency. The test results show that the proposed method is practical for real-time applications. M. Arioli. The use of QR factorization in sparse quadratic programming and backward error issues. SIAM Journal on Matrix Analysis and Applications, 21(3), 825–839, 2000. Abstract. We present a roundoff error analysis of a null space method for solving quadratic programming minimization problems. This method combines the use of a direct QR factorization of the constraints with an iterative solver on the corresponding null space. Numerical experiments are presented which give evidence of the good performances of the algorithm on sparse matrices. We present a roundoff error analysis of a null space method for solving quadratic programming minimization problems. This method combines the use of a direct QR factorization of the constraints with an iterative solver on the corresponding null space. Numerical experiments are presented which give evidence of the good performances of the algorithm on sparse matrices. B. Armstrong and B. A. Wade. Nonlinear pid control with partial state knowledge: design by quadratic programming. In ‘Proceedings of the 2000 American Control Conference, Danvers, MA, USA’, Vol. 2, pp. 774–778, 2000. Abstract. Nonlinear PID (NPID) control is implemented by allowing the controller gains to vary as a function of system state. NPID controllers will in general depend on knowledge of the full state vector. In this work, NPID controllers which operate without knowledge of some state variables are demonstrated. A general but conservative design method is presented with an experimental demonstration. For a special case, complete necessary and sufficient conditions are established. Nonlinear PID (NPID) control is implemented by allowing the controller gains to vary as a function of system state. NPID controllers will in general depend on knowledge of the full state vector. In this work, NPID controllers which operate without knowledge of some state variables are demonstrated. A general but conservative design method is presented with an experimental demonstration. For a special case, complete necessary and sufficient conditions are established. D. A. Arsamastsev, P. I. Bartolomey, and S. K. Okulovski. New improved quadraticprogramming methods for super-large power-systems analysis. In ‘Proceedings of the Eighth Power Systems Computation Conference’, Vol. 178, pp. 710–716, 1984. L. Arseneau and M. J. Best. Resolution of degenerate critical parameter values in parametric quadratic programming. Technical Report CORR 99-47, Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada, 1999. J. Atkociunas. Quadratic programming for degenerate shakedown problems of bar structures. Mechanics Research Communications, 23(2), 195–203, 1996. A. M. Aurela and J. J. Torsti. A quadratic programming method for stabilized solution of unstable linear systems. Annales Universitatis Turkuensis, Ser AI (Astronomica Chemica Physica Mathematica), 123, 1968. N. I. M. GOULD & PH. L. TOINT 7 Abstract. An improved version is presented of the quadratic programming method introduced in 1967 for stabilized solution of unstable systems of linear equations. The most probable solution and its confidence limits are discussed. In the present work, the method proper was examined more systematically by using the second artificial example of Phillips (1962), in which the correct result was known. The dependence of the computing time t on the number of variables adjusted simultaneously, l, was studied, taking a total of m 15 variables, subject to the non negativity constraint. The optimum was achieved with l 3 to 6 (final precision ε 1 105 , t 3 min in the IBM 1130). The different solutions exhibited marked consistency with each other, indicating the accuracy and reliability of the method An improved version is presented of the quadratic programming method introduced in 1967 for stabilized solution of unstable systems of linear equations. The most probable solution and its confidence limits are discussed. In the present work, the method proper was examined more systematically by using the second artificial example of Phillips (1962), in which the correct result was known. The dependence of the computing time t on the number of variables adjusted simultaneously, l, was studied, taking a total of m 15 variables, subject to the non negativity constraint. The optimum was achieved with l 3 to 6 (final precision ε 1 105 , t 3 min in the IBM 1130). The different solutions exhibited marked consistency with each other, indicating the accuracy and reliability of the method G. Auxenfants, L. Barthe, and P. Gibert. Architecture for scientific software. II. Analysis of a quadratic programming algorithm. Recherche Aerospatiale, 4, 247–255, 1982. Abstract. Presents a quadratic programming algorithm with linear constraints, working in the case of largescale optimization problems. The number of variables is reduced by a partial dualization of constraint relations. It enables one to determine whether or not the admissible set is empty. The programming has been implemented on a CYBER 170/750 using a method of architecture based on (1) data centralization and (2) management of information exchange between processors by database management system. This algorithm represents one of the elements of a more optimization code. Presents a quadratic programming algorithm with linear constraints, working in the case of largescale optimization problems. The number of variables is reduced by a partial dualization of constraint relations. It enables one to determine whether or not the admissible set is empty. The programming has been implemented on a CYBER 170/750 using a method of architecture based on (1) data centralization and (2) management of information exchange between processors by database management system. This algorithm represents one of the elements of a more optimization code. T. Aykin. On a quadratic integer-program for the location of interacting hub facilities. European Journal of Operational Research, 46(3), 409–411, 1990. A. Bachem and B. Korte. An algorithm for quadratic optimization over transportation polytopes. Zeitschrift für Angewandte Mathematik und Mechanik, 58, 459–461, 1978. R. Bacher and H. P. van Meeteren. Security dispatch based on coupling of linear and quadratic programming techniques. Power Systems, Modelling and Control Applications. IFAC Symposium Pergamon, Oxford, England, pp. 211–217, 1989. Abstract. Security dispatch can be defined as the real-time closed loop cost-optimal allocation of active generator output while considering branch flow limits of the intact network and lower and upper generation limits. Most OPF algorithms fail to guarantee accuracy. reliability and speed at the same time and can thus not be used in real-time closed loop application. Accuracy, reliability and speed can be obtained by executing a LP based OPF and a QP based constrained economic dispatch at different execution frequencies. The QP based algorithm uses the critical constraint set as determined by the LP based algorithm. Constrained economic dispatch can substitute the classical economic dispatch and will provide a secure dispatch. Security dispatch can be defined as the real-time closed loop cost-optimal allocation of active generator output while considering branch flow limits of the intact network and lower and upper generation limits. Most OPF algorithms fail to guarantee accuracy. reliability and speed at the same time and can thus not be used in real-time closed loop application. Accuracy, reliability and speed can be obtained by executing a LP based OPF and a QP based constrained economic dispatch at different execution frequencies. The QP based algorithm uses the critical constraint set as determined by the LP based algorithm. Constrained economic dispatch can substitute the classical economic dispatch and will provide a secure dispatch. W. E. Baethgen, D. B. Taylor, and M. M. Alley. Quadratic-programming method for determining optimum nitrogen rates for winter-wheat during tillering. Agronomy Journal, 81(4), 557–559, 1989. A. Bagchi and B. Kalantari. A method for computing approximate solution of the trust region problem with application to projective methods for quadratic programming. Working paper, Department of Computer Science, Rutgers University, New Brunswick, New Jersey, USA, 1988. J. R. Baker. Determination of an optimal forecast model for ambulance demand using goal and quadratic programming. In ‘Proceedings—Southeastern Chapter of the Institute of Management Sciences Twentieth Annual Meeting’, Vol. 6, pp. 154–157, 1984. E. Balas. Duality in discrete programming: II. The quadratic case. Management Science, 16, 14–32, 1969. 8 A QUADRATIC PROGRAMMING BIBLIOGRAPHY E. Balas. Nonconvex quadratic programming via generalized polars. SIAM Journal on Applied Mathematics, 28(2), 335–349, 1975. Abstract. A new approach is proposed to linearly constrained nonconvex quadratic programming. The approach is based on generalized polar sets, and is akin to the convex analysis approach to integer programming. The author constructs a generalized polar of the Kuhn-Tucker polyhedron associated with a quadratic program. This generalized polar is a convex polyhedral cone whose interior contains no complementary feasible solution better than the best known one. An algorithm is then proposed, which does not use cutting planes, but constructs a polytope containing the feasible set and contained in the polar of the latter. The best complementary solution found in the process is optimal, or none exists. A new approach is proposed to linearly constrained nonconvex quadratic programming. The approach is based on generalized polar sets, and is akin to the convex analysis approach to integer programming. The author constructs a generalized polar of the Kuhn-Tucker polyhedron associated with a quadratic program. This generalized polar is a convex polyhedral cone whose interior contains no complementary feasible solution better than the best known one. An algorithm is then proposed, which does not use cutting planes, but constructs a polytope containing the feasible set and contained in the polar of the latter. The best complementary solution found in the process is optimal, or none exists. C. C. Baniotopoulos, K. M. Abdalla, and P. D. Panagiotopoulos. A variational inequality and quadratic programming approach to the separation problem of steel bolted brackets. Computers and Structures, 53(4), 983–991, 1994. Abstract. A variational inequality and quadratic programming approach is proposed for the investigation of the separation problem of steel bolted brackets. By applying the classic unilateral contact law to describe the separation process along the contact surfaces between the bracket and the column flange, the continuous problem is formulated as a variational inequality or as a quadratic programming problem. By applying an appropriate finite element discretization scheme, the discrete problem is formulated as a quadratic optimization problem with inequality constraints which, in turn, can be effectively treated numerically by means of an appropriate quadratic optimization algorithm. The applicability and the effectiveness of the method is illustrated by means of a numerical application. A variational inequality and quadratic programming approach is proposed for the investigation of the separation problem of steel bolted brackets. By applying the classic unilateral contact law to describe the separation process along the contact surfaces between the bracket and the column flange, the continuous problem is formulated as a variational inequality or as a quadratic programming problem. By applying an appropriate finite element discretization scheme, the discrete problem is formulated as a quadratic optimization problem with inequality constraints which, in turn, can be effectively treated numerically by means of an appropriate quadratic optimization algorithm. The applicability and the effectiveness of the method is illustrated by means of a numerical application. C. C. Baniotopoulos and K. M. Abdalla. Steel column-to-column connections under combined load—a quadratic-programming approach. Computers and Structures, 46(1), 13– 20, 1993. Abstract. The aim of this paper is the investigation of the mechanical behaviour of bolted steel columnto-column connections under moment and axial loads by means of a method that takes into account the possibility of the appearance of detachment phenomena between the splice plates. As is well known, regions of detachment (called nonactive contact regions below), due to the appearance of the prying-action phenomenon, do appear on the adjacent fronts of such steel splice plates, greatly affecting the mechanical response of steel connections of this type. The significance of the problem under investigation arises from the fact that column-to-column splices are extensively applied in any possible combination to the design and construction of steel structures. It is therefore obvious that, since local failure phenomena on such connections due to undesirable-and not a priori defined-detachment between the splice plates (as consequence of the development of the pryingaction phenomenon) may cause a total destruction of the whole steel structure. For this reason, it is important for such a behaviour to be accurately predicted and the previously mentioned nonactive contact regions on the splice plates to be defined. In this sense, such an investigation leads to an amelioration of the design principles for bolted steel columnto-column splices and to a refinement of the respective steel construction standards. The aim of this paper is the investigation of the mechanical behaviour of bolted steel columnto-column connections under moment and axial loads by means of a method that takes into account the possibility of the appearance of detachment phenomena between the splice plates. As is well known, regions of detachment (called nonactive contact regions below), due to the appearance of the prying-action phenomenon, do appear on the adjacent fronts of such steel splice plates, greatly affecting the mechanical response of steel connections of this type. The significance of the problem under investigation arises from the fact that column-to-column splices are extensively applied in any possible combination to the design and construction of steel structures. It is therefore obvious that, since local failure phenomena on such connections due to undesirable-and not a priori defined-detachment between the splice plates (as consequence of the development of the pryingaction phenomenon) may cause a total destruction of the whole steel structure. For this reason, it is important for such a behaviour to be accurately predicted and the previously mentioned nonactive contact regions on the splice plates to be defined. In this sense, such an investigation leads to an amelioration of the design principles for bolted steel columnto-column splices and to a refinement of the respective steel construction standards. B. Bank and R. Hansel. Stability of mixed-integer quadratic-programming problems. Mathematical Programming Studies, 21, 1–17, 1982. F. Barahona, M. Junger, and G. Reinelt. Experiments in quadratic 0–1 programming. Mathematical Programming, 44(2), 127–137, 1989. E. W. Barankin and R. Dorfman. Toward quadratic programming. Report to the logistics branch, Office of Naval Research, 1955. E. W. Barankin and R. Dorfman. A method for quadratic programming. Econometrica, 24, 1956. N. I. M. GOULD & PH. L. TOINT 9 E. W. Barankin and R. Dorfman. On quadratic programming. University of California Publications in Statistics, 2(13), 285–318, 1958. H. J. C. Barbosa, F. M. P. Raupp, and C. C. H. Borges. Numerical experiments with algorithms for bound constrained quadratic programming in mechanics. Computers and Structures, 64(1–4), 579–594, 1997. Abstract. In this work, the computational performance of some algorithms for solving bound constrained quadratic programming problems is compared by means of numerical experiments. The model problems used to test the behaviour of the algorithms considered were the obstacle problem for a membrane and the contact problem in infinitesimal elasticity. Both problems involved different load conditions and parameters. The finite element method was used for the spatial discretization process. In this work, the computational performance of some algorithms for solving bound constrained quadratic programming problems is compared by means of numerical experiments. The model problems used to test the behaviour of the algorithms considered were the obstacle problem for a membrane and the contact problem in infinitesimal elasticity. Both problems involved different load conditions and parameters. The finite element method was used for the spatial discretization process. J. L. Barlow and G. Toraldo. The effect of diagonal scaling on projected gradient methods for bound constrained quadratic programming problems. Optimization Methods and Software, 5(3), 235–245, 1995. R. O. Barr. An efficient computational procedure for a generalized quadratic programming problem. SIAM Journal on Control, 7(3), 415–429, 1999. R. H. Bartels, G. H. Golub, and M. A. Saunders. Numerical techniques in mathematical programming. In J. B. Rosen, O. L. Mangasarian and K. Ritter, eds, ‘Nonlinear Programming’, pp. 123–176. Academic Press, London, England, 1970. G. Bashein and M. Enns. Computation of optimal controls by a method combining quasilinearization and quadratic programming. International Journal of Control, 16(1), 177– 187, 1972. Abstract. Quadratic programming (QP) has previously been applied to the computation of the optimal controls for linear systems with quadratic cost criteria. This paper extends the application of QP to nonlinear problems through quasi-linearization and the solution of a sequence of linear-quadratic sub-problems whose solutions converge to the solution of the original non-linear problem. The method is called quasilinearizationquadratic programming or Q-QP. Quadratic programming (QP) has previously been applied to the computation of the optimal controls for linear systems with quadratic cost criteria. This paper extends the application of QP to nonlinear problems through quasi-linearization and the solution of a sequence of linear-quadratic sub-problems whose solutions converge to the solution of the original non-linear problem. The method is called quasilinearizationquadratic programming or Q-QP. E. M. L. Beale. On quadratic programming. Naval Research Logistics Quarterly, 6(3), 227– 243, 1959. E. M. L. Beale. The use of quadratic programming in stochastic linear programming. Technical Report P-2404-1, The RAND Corporation, Santa Monica, CA, USA, 1961. E. M. L. Beale. Note on ’a comparison of two methods in quadratic programmng. Operations Research, 14, 442–443, 1966. E. M. L. Beale. An introduction to Beale’s method of quadratic programming. In J. Abadie, ed., ‘Nonlinear programming’, pp. 143–153, North Holland, Amsterdam, the Netherlands, 1967. E. M. L. Beale and R. Benveniste. Quadratic programming. In ‘Design and Implementation of Optimization Software’, pp. 249–258. Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1978. 10 A QUADRATIC PROGRAMMING BIBLIOGRAPHY Abstract. Following a general introduction to the theory of quadratic programming, the paper describes computational aspects of a new algorithm for convex quadratic programming. An essential feature is that the only information needed about the objective function is the gradient direction at successive trial solutions (and the value of the objective function at the final solution). The constraints are handled as in the Reduced Gradient Method. The method is essentially a generalization of the method of Conjugate Gradients. But pure Conjugate Gradients, although finite, require a complete restart whenever the set of active constraints changes. If storage space is available, the algorithm stores additional directions in a way that avoids the need for a complete restart. Following a general introduction to the theory of quadratic programming, the paper describes computational aspects of a new algorithm for convex quadratic programming. An essential feature is that the only information needed about the objective function is the gradient direction at successive trial solutions (and the value of the objective function at the final solution). The constraints are handled as in the Reduced Gradient Method. The method is essentially a generalization of the method of Conjugate Gradients. But pure Conjugate Gradients, although finite, require a complete restart whenever the set of active constraints changes. If storage space is available, the algorithm stores additional directions in a way that avoids the need for a complete restart. J. E. Beasley. Heuristic algorithms for the unconstrained binary quadratic programming problem. Technical report, Department of Mathematics, Imperial College of Science and Technology, London, England, 1998. Abstract. In this paper we consider the unconstrained binary quadratic programming problem. This is the problem of maximising a quadratic objective by suitable choice of binary (zero-one) variables. We present two heuristic algorithms based upon tabu search and simulated annealing for this problem. Computational results are presented for a number of publically available data sets involving up to 2500 variables. An interesting feature of our results is that whilst for most problems tabu search dominates simulated annealing for the very largest problems we consider the converse is true. This paper typifies a ”multiple solution technique, single paper” approach, i.e. an approach that within the same paper presents results for a number of different heuristics applied to the same problem. Issues relating to algorithmic design for such papers are discussed. In this paper we consider the unconstrained binary quadratic programming problem. This is the problem of maximising a quadratic objective by suitable choice of binary (zero-one) variables. We present two heuristic algorithms based upon tabu search and simulated annealing for this problem. Computational results are presented for a number of publically available data sets involving up to 2500 variables. An interesting feature of our results is that whilst for most problems tabu search dominates simulated annealing for the very largest problems we consider the converse is true. This paper typifies a ”multiple solution technique, single paper” approach, i.e. an approach that within the same paper presents results for a number of different heuristics applied to the same problem. Issues relating to algorithmic design for such papers are discussed. C. R. Bector. Indefinite quadratic programming with standard errors in objective. Cahiers du Centre d’Etudes de Recherche Opérationalle, 10, 247–253, 1968. C. R. Bector and M. Dahl. Simplex type finite iterative technique and duality for a special type of pseudo-concave quadratic program,. Cahiers du Centre d’Etudes de Recherche Opérationalle, 16, 207–222, 1974. L. Behjat and A. Vannelli. VLSI concentric partitioning using interior point quadratic programming. In ‘ISCAS’99. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSI. IEEE, Piscataway, NJ, USA’, Vol. 6, pp. 93–96, 1999. Abstract. This paper presents a novel approach for solving the standard cell placement problem. A relaxed quadratic formulation of the problem is solved iteratively incorporating techniques to increase the spreading of cells, including introducing attractors and dynamic first moment constraints. At each iteration, a percentage of the cells that are close to the boundary of the chip are fixed. This procedure is done recursively until at least eighty percent of the cells are fixed. Numerical simulation of the proposed approach is presented for test systems. This paper presents a novel approach for solving the standard cell placement problem. A relaxed quadratic formulation of the problem is solved iteratively incorporating techniques to increase the spreading of cells, including introducing attractors and dynamic first moment constraints. At each iteration, a percentage of the cells that are close to the boundary of the chip are fixed. This procedure is done recursively until at least eighty percent of the cells are fixed. Numerical simulation of the proposed approach is presented for test systems. L. Y. Belousov. Quadratic programming in problems of optimal planning of trajectory measurements. Cosmic Research, 9(6), 750–759, 1971. Abstract. The problem of optimal planning of trajectory measurements of two different measured parameters is investigated for the case of a limited dispersion, an arbitrary correlation coupling of the measurement errors of each parameter separately, and absence of correlation between measurements of different parameters. It is shown that the problem posed can be reduced to solving a problem in quadratic programming based on the linear-programming method generalized for the continuous case. In conclusion, the problem is stated by induction for an arbitrary number of independent measured parameters. The problem of optimal planning of trajectory measurements of two different measured parameters is investigated for the case of a limited dispersion, an arbitrary correlation coupling of the measurement errors of each parameter separately, and absence of correlation between measurements of different parameters. It is shown that the problem posed can be reduced to solving a problem in quadratic programming based on the linear-programming method generalized for the continuous case. In conclusion, the problem is stated by induction for an arbitrary number of independent measured parameters. T. Belytschko. Discussion of elastic-plastic analysis by quadratic programming. American Society of Civil Engineering, Journal of the Engineering Mechanics Division, 100, 130– 131, 1974. N. I. M. GOULD & PH. L. TOINT 11 A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos. The explicit solution of model predictive control via multiparametric quadratic programming. In ‘Proceedings of the 2000 American Control Conference, Danvers, MA, USA’, Vol. 2, pp. 872–876, 2000. Abstract. The control based on online optimization, popularly known as model predictive control (MPC), has long been recognized as the winning alternative for constrained systems. The main limitation of MPC is, however, its online computational complexity. For discrete-time linear time-invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly the state feedback control law associated with MPC, and show that it is piecewise linear and continuous. The controller inherits all the stability and performance properties of MPC, but the online computation is reduced to a simple linear function evaluation instead of the expensive quadratic program. The new technique is expected to enlarge the scope of applicability of MPC to small-size/fast-sampling applications which cannot be covered satisfactorily with anti-windup schemes. The control based on online optimization, popularly known as model predictive control (MPC), has long been recognized as the winning alternative for constrained systems. The main limitation of MPC is, however, its online computational complexity. For discrete-time linear time-invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly the state feedback control law associated with MPC, and show that it is piecewise linear and continuous. The controller inherits all the stability and performance properties of MPC, but the online computation is reduced to a simple linear function evaluation instead of the expensive quadratic program. The new technique is expected to enlarge the scope of applicability of MPC to small-size/fast-sampling applications which cannot be covered satisfactorily with anti-windup schemes. M. Ben Daya. Line search techniques for the logarithmic barrier function in quadratic programming. Journal of the Operational Research Society, 46(3), 332–338, 1995. Abstract. In this paper, we propose a line-search procedure for the logarithmic barrier function in the context of an interior point algorithm for convex quadratic programming. Preliminary testing shows that the proposed procedure is superior to some other line-search methods developed specifically for the logarithmic barrier function in the literature. In this paper, we propose a line-search procedure for the logarithmic barrier function in the context of an interior point algorithm for convex quadratic programming. Preliminary testing shows that the proposed procedure is superior to some other line-search methods developed specifically for the logarithmic barrier function in the literature. M. Ben Daya and K. S. Al Sultan. A new penalty function algorithm for convex quadratic programming. European Journal of Operational Research, 101(1), 155–163, 1997. Abstract. We develop an exterior point algorithm for convex quadratic programming using a penalty function approach. Each iteration in the algorithm consists of a single Newton step followed by a reduction in the value of the penalty parameter. The points generated by the algorithm follow an exterior path that we define. Convergence of the algorithm is established. The proposed algorithm was motivated by the work of Al-Sultan and Murty (1991) on nearest point problems, a special quadratic program. A preliminary implementation of the algorithm produced encouraging results. In particular, the algorithm requires a small and almost constant number of iterations to solve the small to medium size problems tested. We develop an exterior point algorithm for convex quadratic programming using a penalty function approach. Each iteration in the algorithm consists of a single Newton step followed by a reduction in the value of the penalty parameter. The points generated by the algorithm follow an exterior path that we define. Convergence of the algorithm is established. The proposed algorithm was motivated by the work of Al-Sultan and Murty (1991) on nearest point problems, a special quadratic program. A preliminary implementation of the algorithm produced encouraging results. In particular, the algorithm requires a small and almost constant number of iterations to solve the small to medium size problems tested. M. Ben Daya and C. M. Shetty. Polynomial barrier function algorithms for convex quadratic programming. Arabian Journal for Science and Engineering, 15(4B), 656–670, 1990. J. M. Bennett. Quadratic programming and piecewise linear networks with structural engineering applications,. In A. Prekopa, ed., ‘Survey of Mathematical Programming, Vol. 3’, pp. 95–105, North Holland, Amsterdam, the Netherlands, 1979. P. Benson, R. L. Smith, I. E. Schochetman, and J. C. Bean. Optimal solution approximation for infinite positive-definite quadratic programming. Journal of Optimization Theory and Applications, 85(2), 235–248, 1995. Abstract. We consider a general doubly-infinite, positive-definite, quadratic programming problem. We show that the sequence of unique optimal solutions to the natural finite-dimensional subproblems strongly converges to the unique optimal solution. This offers the opportunity to arbitrarily well approximate the infinite-dimensional optimal solution by numerically solving a sufficiently large finite-dimensional version of the problem. We then apply our results to a general time-varying, infinitehorizon, positive-definite, LQ control problem. We consider a general doubly-infinite, positive-definite, quadratic programming problem. We show that the sequence of unique optimal solutions to the natural finite-dimensional subproblems strongly converges to the unique optimal solution. This offers the opportunity to arbitrarily well approximate the infinite-dimensional optimal solution by numerically solving a sufficiently large finite-dimensional version of the problem. We then apply our results to a general time-varying, infinitehorizon, positive-definite, LQ control problem. R. Benveniste. A quadratic programming algorithm using conjugate search directions. Mathematical Programming, 16(1), 63–80, 1979. 12 A QUADRATIC PROGRAMMING BIBLIOGRAPHY Abstract. A quadratic programming algorithm is presented, resembling Beale’s (1955) quadratic programming algorithm and Wolfe’s reduced gradient method. It uses conjugate search directions. The algorithm is conceived as being particularly appropriate for problems with a large Hessian matrix. An outline of the solution to the quadratic capacity-constrained transportation problem using the above method is also presented. A quadratic programming algorithm is presented, resembling Beale’s (1955) quadratic programming algorithm and Wolfe’s reduced gradient method. It uses conjugate search directions. The algorithm is conceived as being particularly appropriate for problems with a large Hessian matrix. An outline of the solution to the quadratic capacity-constrained transportation problem using the above method is also presented. R. Benveniste. One way to solve the parametric quadratic programming problem. Mathematical Programming, 21(2), 224–228, 1981. Abstract. A method is presented for the solution of the parametric quadratic programming problem by the use of conjugate directions. A method is presented for the solution of the parametric quadratic programming problem by the use of conjugate directions. C. Bergthaller. A quadratic equivalent of the minimum risk problem. Revue Roumaine do Mathematiques Pure et Appliquees, 15, 17–23, 1970. C. Bergthaller. Parametric quadratic programming. In ‘4th conference on probability theory. Abstracts. Acad Socialist Republic of Rumania, Bucharest, Romania’, pp. 23–24, 1971a. Abstract. This paper deals with the parametric programming problem mincT x 1 2xT Dx A λ x b, x 0, where: c x Rn, b Rm, D is a symmetric nxn positive definite matrix, A λ Ao λAl , Ao and Al are fixed m n matrices, such that the rank of Al is l and λ 0 is real parameter. Some particular cases are: 1) One element of the matrix A is a linear function of lambda and all others are constant. 2) A column of mod A is a linear (vectorial) function of λ a j aj λaj and the others are constant. 3) A row of A is a linear (vectorial) function of λ α j αj λαi and the others are constant. C. Bergthaller. Quasi-convex quadratic programming. Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences, Serie A (Science Mathematiques), 273(15), 685– 686, 1971b. Abstract. A simplicial algorithm is given for the program minqT x 1 2xT Qx Ax b x 0 where q is an n-dimensional vector and Q is a symmetrical matrix such that the objective function f x identical to qT x 1 2xT Qx is quasi-convex for x 0 without being convex. A. B. Berkelaar. Sensitivity analysis in (degenerate) quadratic programming. Econometric Institute Report 30, Econometric Institute, Erasmus University, Rotterdam, The Netherlands, 1997. Abstract. In this paper we deal with sensitivity analysis in convex quadratic programming, without making assumptions on nondegeneracy, strict convexity of the objective function, and the existence of a strictly complementary solution. We show that the optimal value as a function of a right–hand side element (or an element of the linear part of the objective) is piecewise quadratic, where the pieces can be characterized by maximal complementary solutions and tripartitions. Further, we investigate differentiability of this function. A new algorithm to compute the optimal value function is proposed. Finally, we discuss the advantages of this approach when applied to mean–variance portfolio models. In this paper we deal with sensitivity analysis in convex quadratic programming, without making assumptions on nondegeneracy, strict convexity of the objective function, and the existence of a strictly complementary solution. We show that the optimal value as a function of a right–hand side element (or an element of the linear part of the objective) is piecewise quadratic, where the pieces can be characterized by maximal complementary solutions and tripartitions. Further, we investigate differentiability of this function. A new algorithm to compute the optimal value function is proposed. Finally, we discuss the advantages of this approach when applied to mean–variance portfolio models. A. B. Berkelaar, B. Jansen, C. Roos, and T. Terlaky. Sensitivity analysis in quadratic programming. Report 96-11, Econometric Institute, Erasmus University, Rotterdam, The Netherlands, 1996. Abstract. In this paper we deal with sensitivity analysis in convex quadratic programming, without making assumptions on nondegeneracy, strict convexity of the objective function, and the existence of a strictly complementary solution. We show that the optimal value as a function of a right–hand side element (or an element of the linear part of the objective) is piecewise quadratic, where the pieces can be characterized by maximal complementary solutions and tripartitions. Further, we investigate differentiability of this function. A new algorithm to compute the optimal value function is proposed. Finally, we discuss the advantages of this approach when applied to mean–variance portfolio models. In this paper we deal with sensitivity analysis in convex quadratic programming, without making assumptions on nondegeneracy, strict convexity of the objective function, and the existence of a strictly complementary solution. We show that the optimal value as a function of a right–hand side element (or an element of the linear part of the objective) is piecewise quadratic, where the pieces can be characterized by maximal complementary solutions and tripartitions. Further, we investigate differentiability of this function. A new algorithm to compute the optimal value function is proposed. Finally, we discuss the advantages of this approach when applied to mean–variance portfolio models. N. I. M. GOULD & PH. L. TOINT 13 A. B. Berkelaar, B. Jansen, K. Roos, and T. Terlaky. Basisand partition identification for quadratic programming and linear complementarity problems. Mathematical Programming, 86(2), 261–282, 1999. Abstract. Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally complementary solutions. Maximally complementary solutions can be characterized by optimal partitions. On the other hand, the solutions provided by simplex-based pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A partition identification algorithm is an algorithm which generates a maximally complementary solution (and its corresponding partition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus. Optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally complementary solutions. Maximally complementary solutions can be characterized by optimal partitions. On the other hand, the solutions provided by simplex-based pivot algorithms are given in terms of complementary bases. A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution. A partition identification algorithm is an algorithm which generates a maximally complementary solution (and its corresponding partition), starting from any complementary solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus. A. B. Berkelaar, K. Roos, and T. Terlaky. The optimal partition and optimal set approach to linear and quadratic programming. Econometric Institute Report 51, Econometric Institute, Erasmus University, Rotterdam, The Netherlands, 1997. Abstract. In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transition-points of the optimal value function. The advantage of using this approach instead of the classical approach (using optimal bases) is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss differences and resemblances with the linear programming case. In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transition-points of the optimal value function. The advantage of using this approach instead of the classical approach (using optimal bases) is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss differences and resemblances with the linear programming case. H. Bernau. Upper bound techniques for quadratic programming. Alkalmazott Matematikai Lapok, 3(1–2), 161–170, 1977. See also, A. Prekopa, ed. Survey of Mathematical Programming, Vol.1, North-Holland, Amsterdam, pp. 347–356, 1979. Abstract. An extension of the methods of Wolfe, Jagannathan and Beale is presented for quadratic programming problems with upper bounds for the variables. It is shown that the upper bounds technique for linear programming problems can be very easily incorporated in these methods. An extension of the methods of Wolfe, Jagannathan and Beale is presented for quadratic programming problems with upper bounds for the variables. It is shown that the upper bounds technique for linear programming problems can be very easily incorporated in these methods. H. Bernau. Quadratic programming problems and related linear complementarity problems. Journal of Optimization Theory and Applications, 65(2), 209–222, 1990. Abstract. Investigates the general quadratic programming problem, i.e. the problem of finding the minimum of a quadratic function subject to linear constraints. In the case where, over the set of feasible points, the objective function is bounded from below, this problem can be solved by the minimization of a linear function, subject to the solution set of a linear complementarity problem, representing the Kuhn-Tucker conditions of the quadratic problem. To detect in the quadratic problem the unboundedness from below of the objective function, necessary and sufficient conditions are derived. It is shown that, when these conditions are applied, the general quadratic programming problem becomes equivalent to the investigation of an appropriately formulated linear complementary problem. Investigates the general quadratic programming problem, i.e. the problem of finding the minimum of a quadratic function subject to linear constraints. In the case where, over the set of feasible points, the objective function is bounded from below, this problem can be solved by the minimization of a linear function, subject to the solution set of a linear complementarity problem, representing the Kuhn-Tucker conditions of the quadratic problem. To detect in the quadratic problem the unboundedness from below of the objective function, necessary and sufficient conditions are derived. It is shown that, when these conditions are applied, the general quadratic programming problem becomes equivalent to the investigation of an appropriately formulated linear complementary problem. O. Bertoldi, M. V. Cazzol, A. Garzillo, and M. Innorta. A dual quadratic programming algorithm oriented to the probabilistic analysis of large interconnected networks. In ‘PSCC. Proceedings of the Twelfth Power Systems Computation Conference. Power Syst. Comput. Conference, Zurich, Switzerland’, Vol. 2, pp. 1249–1255, 1996. Abstract. A fast and robust innovative computing procedure has been developed aimed at allowing the use of optimal power flow techniques in the framework of the probabilistic adequacy assessment of large interconnected power systems. The paper describes the methodological approach and the relevant implemented A fast and robust innovative computing procedure has been developed aimed at allowing the use of optimal power flow techniques in the framework of the probabilistic adequacy assessment of large interconnected power systems. The paper describes the methodological approach and the relevant implemented 14 A QUADRATIC PROGRAMMING BIBLIOGRAPHY algorithm. Several numerical results are supplied which demonstrate the high computing efficiency of the procedure so that it is suitable in the probabilistic simulation domain. M. J. Best. Equivalence of some quadratic programming algorithms. Mathematical Programming, 30(1), 71–87, 1984. Abstract. The author formulates a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher (1971), Gill and Murray (1978), Best and Ritter (1976), and van de Panne and Whinston/Dantzig (1969) are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb (1972) and Djang (1979) for the positive definite Hessian case. The analysis extends these results to the positive semi-definite case. The author formulates a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher (1971), Gill and Murray (1978), Best and Ritter (1976), and van de Panne and Whinston/Dantzig (1969) are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb (1972) and Djang (1979) for the positive definite Hessian case. The analysis extends these results to the positive semi-definite case. M. J. Best. An algorithm for parametric quadratic programming. In H. Fischer, B. Riedüller and S. Schäffler, eds, ‘Applied Mathematics and Parallel Computing—Festschrift for Klaus Ritter’, pp. 57–76. Physica-Verlag, Heidelburg, 1996. M. J. Best and R. J. Caron. A method to increase the computational efficiency of certain quadratic programming algorithms. Mathematical Programming, 25(3), 354–358, 1983. Abstract. Presents a method for computing the Kuhn-Tucker multipliers associated with equality constraints in quadratic programming problems. When applied to a certain class of algorithms a significant reduction in computation time and in storage is achieved. Presents a method for computing the Kuhn-Tucker multipliers associated with equality constraints in quadratic programming problems. When applied to a certain class of algorithms a significant reduction in computation time and in storage is achieved. M. J. Best and R. J. Caron. A parameterized Hessian quadratic programming problem. Annals of Operations Research, 5(1–4), 373–394, 1986. Abstract. Presents a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem. Presents a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem. M. J. Best and N. Chakravarti. Stability of linearly constrained convex quadratic programs. Journal of Optimization Theory and Applications, 64(1), 43–53, 1990. M. J. Best and N. Chakravarti. An O n2 active set method for solving a certain parametric quadratic program. Journal of Optimization Theory and Applications, 72(2), 213–224, 1992. Abstract. The paper presents an O n2 method for solving the parametric quadratic program min 1 2 xT Dx aT x λ 2 ∑j 1 γ jx j c 2, having lower and upper bounds on the variables, for all nonnegative values of the parameter lambda . Here, D is a positive diagonal matrix, a an arbitrary n-vector, each γ j , j 1 n, and c are arbitrary scalars. An application to economics is also presented. The paper presents an O n2 method for solving the parametric quadratic program min 1 2 xT Dx aT x λ 2 ∑j 1 γ jx j c 2, having lower and upper bounds on the variables, for all nonnegative values of the parameter lambda . Here, D is a positive diagonal matrix, a an arbitrary n-vector, each γ j , j 1 n, and c are arbitrary scalars. An application to economics is also presented. M. J. Best and K. Ritter. An effective algorithm for quadratic minimization problems. Technical report 1691, University of Wisconsin, Madison, Wisconsin, USA, 1976. M. J. Best and K. Ritter. A quadratic programming algorithm. ZOR, Methods and Models of Operations Research, 32(5), 271–297, 1988. N. I. M. GOULD & PH. L. TOINT 15 Abstract. By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed. By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed. D. Bhatia. Duality for quadratic programming in complex space. Zeitschrift für Angewandte Mathematik und Mechanik, 54(1), 55–57, 1974. Abstract. The duality theorems of Rami (1972) for symmetric dual quadratic programs are extended in complex space over arbitrary polyhedral cones. The duality theorems of Rami (1972) for symmetric dual quadratic programs are extended in complex space over arbitrary polyhedral cones. S. Bhowmik, S. K. Goswami, and P. K. Bhattacherjee. Distribution system planning through combined heuristic and quadratic programming approach. Electric Machines and Power Systems, 28(1), 87–103, 2000. Abstract. The present paper reports a new technique for the planning of a radial distribution system. Distribution system planning has been formulated as a problem of quadratic mixed integer programming (QMIP). A two-stage iterative solution technique has been proposed where the first stage determines the optimum substation sites and the second stage determines the optimum network configurations. To reduce the dimensionality problem, the integer constraints are first related, thus converting the quadratic mixed integer programming problem into a quadratic programming (QP) problem. After the solution of the QP problem, integer constraints are imposed using heuristic techniques. The present paper reports a new technique for the planning of a radial distribution system. Distribution system planning has been formulated as a problem of quadratic mixed integer programming (QMIP). A two-stage iterative solution technique has been proposed where the first stage determines the optimum substation sites and the second stage determines the optimum network configurations. To reduce the dimensionality problem, the integer constraints are first related, thus converting the quadratic mixed integer programming problem into a quadratic programming (QP) problem. After the solution of the QP problem, integer constraints are imposed using heuristic techniques. D. Bienstock. Computational study of a family of mixed-integer quadratic programming problems. Mathematical Programming, 74(2), 121–140, 1996. See also, Integer Programming and Combinatorial Optimization. 4th International IPOC Conference, Proceedings (Balas, E. and Clausen, J., eds.), Springer-Verlag, Berlin, Germany, pages 90–94, 1995. Abstract. We present computational experience with a branch-and-cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest real-life problems in a few minutes of run-time. We present computational experience with a branch-and-cut algorithm to solve quadratic programming problems where there is an upper bound on the number of positive variables. Such problems arise in financial applications. The algorithm solves the largest real-life problems in a few minutes of run-time. A. Billionnet and A. Sutter. Persistency in quadratic 0–1 optimization. Mathematical Programming, 54(1), 115–119, 1992. Å. Björck. Constrained least-squares problems. In ‘Numerical Methods for Least Squares Problems’, chapter 5, pp. 187–213. SIAM, Philadelphia, USA, 1996. E. Blum and W. Oettli. Direct proof of the existence theorem for quadratic programming. Operations Research, 20(1), 165–167, 1972. Abstract. A direct analytical proof is given for the following theorem: If the infimum of a quadratic function on a nonempty (possibly unbounded) polyhedral set R contained in IRn is finite, then the infimum is assumed somewhere on R, thus being a minimum. A direct analytical proof is given for the following theorem: If the infimum of a quadratic function on a nonempty (possibly unbounded) polyhedral set R contained in IRn is finite, then the infimum is assumed somewhere on R, thus being a minimum. P. T. Boggs, P. D. Domich, and J. E. Rogers. An interior point method for general large-scale quadratic programming problems. Annals of Operations Research, 62, 419–437, 1996a. Abstract. Presents an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of the authors‘ interior point work on linear programming problems, efficiently solves a wide class of large-scale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a threedimensional cost improvement subproblem, which is solved at every iteration. The authors have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the success of the algorithm. The authors describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented. Presents an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of the authors‘ interior point work on linear programming problems, efficiently solves a wide class of large-scale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a threedimensional cost improvement subproblem, which is solved at every iteration. The authors have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the success of the algorithm. The authors describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented. 16 A QUADRATIC PROGRAMMING BIBLIOGRAPHY P. T. Boggs, P. D. Domich, J. E. Rogers, and C. Witzgall. An interior-point method for linear and quadratic programming problems. COAL Newsletter, 19, 32–40, 1991. P. T. Boggs, P. D. Domich, J. E. Rogers, and C. Witzgall. An interior point method for general large scale quadratic programming problems. Annals of Operations Research, 62, 419– 437, 1996b. Abstract. In this paper we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems, efficiently solves a wide class of large scale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a 3-dimensional cost-improvement subproblem, which is solved at every iteration. We have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the success. We describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented. In this paper we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems, efficiently solves a wide class of large scale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a 3-dimensional cost-improvement subproblem, which is solved at every iteration. We have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the success. We describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented. N. L. Boland. A dual-active-set algorithm for positive semidefinite quadratic programming. In ‘Optimization : Techniques and Applications’, pp. 80–89, 1992. N. L. Boland. A dual-active-set algorithm for positive semi-definite quadratic programming. Mathematical Programming, 78(1), 1–27, 1997. Abstract. Because of the many important applications of quadratic programming, fast and efficient methods for solving quadratic programming problems are valued. Goldfarb and Idnani (1983) describe one such method, Well known to be efficient and numerically stable, the Goldfarb and Idnani method suffers only from the restriction that in its original form it cannot be applied to problems which are positive semi-definite rather than positive definite. In this paper, we present a generalization of the Goldfarb and Idnani method to the positive semi-definite case and prove finite termination of the generalized algorithm. In our generalization, we preserve the spirit of the Goldfarb and Idnani method, and extend their numerically stable implementation in a natural way. Because of the many important applications of quadratic programming, fast and efficient methods for solving quadratic programming problems are valued. Goldfarb and Idnani (1983) describe one such method, Well known to be efficient and numerically stable, the Goldfarb and Idnani method suffers only from the restriction that in its original form it cannot be applied to problems which are positive semi-definite rather than positive definite. In this paper, we present a generalization of the Goldfarb and Idnani method to the positive semi-definite case and prove finite termination of the generalized algorithm. In our generalization, we preserve the spirit of the Goldfarb and Idnani method, and extend their numerically stable implementation in a natural way. I. M. Bomze and G. Danninger. A global optimization algorithm for concave quadratic programming problems. SIAM Journal on Optimization, 3(4), 826–842, 1993. J. F. Bonnans and M. Bouhtov. The trust region affine interior point algorithm for convex and nonconvex quadratic programming. RAIRO-Recherche Opérationnelle—Operations Research, 29(2), 195–217, 1995. Abstract. We study from a theoretical and numerical point of view an interior point algorithm for quadratic QP using a trust region idea, formulated by Ye and Tse (1989). We show that, under a nondegeneracy hypothesis, the algorithm converges globally in the convex case. For a nonconvex problem, under a mild additional hypothesis, the sequence of points converges to a stationary point. We obtain also an asymptotic linear convergence rate for the cost that depends only on the dimension of the problem. When we show that, provided some modifications are added to the basic algorithm, the method has a good numerical behaviour. We study from a theoretical and numerical point of view an interior point algorithm for quadratic QP using a trust region idea, formulated by Ye and Tse (1989). We show that, under a nondegeneracy hypothesis, the algorithm converges globally in the convex case. For a nonconvex problem, under a mild additional hypothesis, the sequence of points converges to a stationary point. We obtain also an asymptotic linear convergence rate for the cost that depends only on the dimension of the problem. When we show that, provided some modifications are added to the basic algorithm, the method has a good numerical behaviour. J. C. G. Boot. Notes on quadratic programming: The Kuhn-Tucker and Thiel-Van de Panne conditions, degeneracy and equality constraints. Management Science, 8, 85–98, 1961. J. C. G. Boot. Binding constraint procedures of quadratic programming. Econometrica, 31, 464–498, 1963a. J. C. G. Boot. On sensitivity analysis in convex quadratic programming problems. Operations Research, 11, 771–786, 1963b. N. I. M. GOULD & PH. L. TOINT 17 J. C. G. Boot. Quadratic Programming. Algorithms—Anomalies—Algorithms. North Holland, Amsterdam, the Netherlands, 1964. J. C. G. Boot and H. Theil. A procedure for integer maximization of a definite quadratic form. In G. Kreweras and G. Morlat, eds, ‘Proceedings of the 3rd International Conference on Operations Research’. English Universities Press, 1964. V. L. Borre and S. G. Kapoor. A multi-stage quadratic-programming optimization technique for optimal management of large hydro-power operations. Engineering Optimization, 8(2), 103–118, 1985. J. M. Borwein. Necessary and sufficient conditions for quadratic minimality. Numerical Functional Analysis and Optimization, 5, 127–140, 1982. R. J. Bosch, Y. Ye, and G. G. Woodworth. A convergent algorithm for quantile regression with smoothing splines. Computational Statistics and Data Analysis, 19, 613–630, 1995. Abstract. An important practical problem is that of determining a nonparametric estimate of the conditional quantile of y given x. If we balance fidelity to the data with a smoothness requirement, the resulting quantile function is a cubic smoothing spline. We reformulate this estimation procedure as a quadratic programming problem, with associated optimality conditions. A recently developed interior point algorithm with proven convergence is extended to solve the quadratic program. This solution characterizes the desired nonparametric conditional quantile function. These methods are illustrated in a study of audiologic performance following cochlear implants. An important practical problem is that of determining a nonparametric estimate of the conditional quantile of y given x. If we balance fidelity to the data with a smoothness requirement, the resulting quantile function is a cubic smoothing spline. We reformulate this estimation procedure as a quadratic programming problem, with associated optimality conditions. A recently developed interior point algorithm with proven convergence is extended to solve the quadratic program. This solution characterizes the desired nonparametric conditional quantile function. These methods are illustrated in a study of audiologic performance following cochlear implants. A. Bouzaher. Symmetric QP and linear programming under primal-dual uncertainty. Operations Research Letters, 6(5), 221–225, 1987. Abstract. A saddle-point formulation of linear programming problems with random objective function and RHS coefficients is proposed. Under a certainty equivalent criterion, a pair of primal-dual deterministic equivalents is derived. These problems are symmetric dual quadratic programs, and can be interpreted as generalizations of the classical mean-variance model. A saddle-point formulation of linear programming problems with random objective function and RHS coefficients is proposed. Under a certainty equivalent criterion, a pair of primal-dual deterministic equivalents is derived. These problems are symmetric dual quadratic programs, and can be interpreted as generalizations of the classical mean-variance model. J. Bouzitat. On Wolfe’s method and Dantzig’s method for convex quadratic programming. RAIRO Recherche Operationelle, 13(2), 151–184, 1979. Abstract. Both Wolfe’s and Dantzig’s methods solve linear-constrained convex quadratic programming problems by simplex-like algorithms. They use the Kuhn-Tucker conditions, which are necessary and sufficient for such problems. The author presents those two methods, with complete theoretical proofs the greater part of which, to the author’s knowledge, is new. Wolfe’s method receives here a complement and is then found to be more efficient than it previously appeared, on account of ’blocking’ phenomena which are proved not to stop the convergent process of the completed algorithm. The ’shear form’ of the method is consequently applicable to solve any nonparametric problem, and the ’long form’ may be reserved for parametric problems only. The present proof of Dantzig’s algorithm convergence is not based on the direct study of computation schemata, but uses the convexity of the quadratic function to be minimized, which leads to quite simple proof. The general presentation of Wolfe’s and Dantzig’s methods is illustrated by a numerical problem which is solved by both of them, so as to permit a comparison. Both Wolfe’s and Dantzig’s methods solve linear-constrained convex quadratic programming problems by simplex-like algorithms. They use the Kuhn-Tucker conditions, which are necessary and sufficient for such problems. The author presents those two methods, with complete theoretical proofs the greater part of which, to the author’s knowledge, is new. Wolfe’s method receives here a complement and is then found to be more efficient than it previously appeared, on account of ’blocking’ phenomena which are proved not to stop the convergent process of the completed algorithm. The ’shear form’ of the method is consequently applicable to solve any nonparametric problem, and the ’long form’ may be reserved for parametric problems only. The present proof of Dantzig’s algorithm convergence is not based on the direct study of computation schemata, but uses the convexity of the quadratic function to be minimized, which leads to quite simple proof. The general presentation of Wolfe’s and Dantzig’s methods is illustrated by a numerical problem which is solved by both of them, so as to permit a comparison. R. J. Braitsch. A computer comparison of four quadratic programming algorithms. Management Science, 18(11), 632–643, 1972. Abstract. This paper compares the computational performance of four quadratic programming algorithms. Problems are generated and solved on the computer with iteration count serving as the principal method of comparison. The effect of certain problem parameters on rate of convergence is considered and computer time and storage requirements of the four algorithms are discussed. This paper compares the computational performance of four quadratic programming algorithms. Problems are generated and solved on the computer with iteration count serving as the principal method of comparison. The effect of certain problem parameters on rate of convergence is considered and computer time and storage requirements of the four algorithms are discussed. 18 A QUADRATIC PROGRAMMING BIBLIOGRAPHY N. J. Breytenbach. A structured quadratic program. In J. A. Snyman, ed., ‘Proceedings of the Tenth South African Symposium on Numerical Mathematics’, p. 128, 1984. J. F. Brotchie. A generalized design approach to solution of the nonconvex quadraticprogramming problem. Applied Mathematical Modelling, 11(4), 291–295, 1987. B. M. Brown and C. R. Goodall. Applications of quadratic programming in statistics. Technical Report 93-09, Department of Statistsics, The Pennsylvania State University, USA, 1993. K. S. Brown and K. A. Reiman. An ALGOL program for quadratic programming using the method of Wolfe and Frank. Bulletin of the Operations Research Society of America, 23, B374, 1975. Abstract. To meet the needs of making available computer programs, this paper presents an ALGOL program for applying the method of Wolfe and Frank to properly structured quadratic programming problems. The procedure is then demonstrated by solving a representative sample of structured problems. Where feasible, problems are formulated in such a way that relevance to real problems is readily noted. To meet the needs of making available computer programs, this paper presents an ALGOL program for applying the method of Wolfe and Frank to properly structured quadratic programming problems. The procedure is then demonstrated by solving a representative sample of structured problems. Where feasible, problems are formulated in such a way that relevance to real problems is readily noted. C. G. Broyden and N. F. Attia. Penalty functions, Newton’s method, and quadratic programming. Journal of Optimization Theory And Applications, 58(3), 377–385, 1988. J. R. Bunch and L. Kaufman. A computational method for the indefinite quadratic programming problem. Linear Algebra and Its Applications, 34, 341–370, 1980. Abstract. Presents an algorithm for the quadratic programming problem of determining a local minimum of f x 1 2xT Qx cT x such that AT x b where Q is a symmetric matrix which may not be positive definite. C. A. Burdet. General quadratic programming. Technical report, Carnegie Mellon Univ , Pittsburgh, PA, USA, 1971. Presents an algorithm for the quadratic programming problem of determining a local minimum of f x 1 2xT Qx cT x such that AT x b where Q is a symmetric matrix which may not be positive definite. C. A. Burdet. General quadratic programming. Technical report, Carnegie Mellon Univ , Pittsburgh, PA, USA, 1971. Abstract. An algorithm is presented for the general (not necessarily convex or concave) quadratic programming problem over a linearly constrained set. The algorithm is finitely convergent and makes use of a convex quadratic programming method as a subroutine (like the quadratic simplex for instance). The basic tool for this method is a facial decomposition for polyhedral sets. An algorithm is presented for the general (not necessarily convex or concave) quadratic programming problem over a linearly constrained set. The algorithm is finitely convergent and makes use of a convex quadratic programming method as a subroutine (like the quadratic simplex for instance). The basic tool for this method is a facial decomposition for polyhedral sets. S. J. Byrne. Solution of quadratic programming problems. New Zealand Operational Research, 12(2), 73–89, 1984. Abstract. Quadratic programming problems arise in a number of situations. Typical examples, e.g. portfolio selection, economic modelling, regression analysis, and solution of non-linear programming problems, are briefly described. The methods that have been proposed for solving this problem are reviewed. The approach using a linear complementarity program is selected, and an efficient, numerically well-behaved procedure is developed, based on Lemke’s algorithm and using orthogonal factorizations. A principal pivot version is also presented, which parallels Dantzig and Cottle’s solution procedure, but is applicable to the same very wide class of matrices processed by Lemke’s algorithm. A restart facility is developed for this version, which accelerates the solution of a sequence of related quadratic programs, by proceeding from the optimum of the last problem. The methods have been coded in Fortran IV and perform well. The process is easily extended to handle both upper and lower bounds on constraints and has proved acceptable for use in a general non-linear programming algorithm. Quadratic programming problems arise in a number of situations. Typical examples, e.g. portfolio selection, economic modelling, regression analysis, and solution of non-linear programming problems, are briefly described. The methods that have been proposed for solving this problem are reviewed. The approach using a linear complementarity program is selected, and an efficient, numerically well-behaved procedure is developed, based on Lemke’s algorithm and using orthogonal factorizations. A principal pivot version is also presented, which parallels Dantzig and Cottle’s solution procedure, but is applicable to the same very wide class of matrices processed by Lemke’s algorithm. A restart facility is developed for this version, which accelerates the solution of a sequence of related quadratic programs, by proceeding from the optimum of the last problem. The methods have been coded in Fortran IV and perform well. The process is easily extended to handle both upper and lower bounds on constraints and has proved acceptable for use in a general non-linear programming algorithm. R. Caballero and A. Santos. A new dual method for solving strictly convex quadratic programs. Technical report, Department of Applied Economics (Mathematics), University of Malaga, Spain, 1998. N. I. M. GOULD & PH. L. TOINT 19 A. V. Cabot and R. L. Francis. Solving nonconvex quadratic minimization problems by ranking the extreme points. Operations Research, 18, 82–86, 1970. L. M. Cabral. An efficient algorithm for the quadratic programming problem with inequality constraints. In ‘Proceedings of the 1982 American Control Conference. IEEE, New York, NY, USA’, Vol. 3, pp. 1016–1017, 1982. Abstract. The solution to the general linear problem Ax y with side conditions can be interpreted as a quadratic optimization problem. Such a requirement arises in the solution of illposed problems where the method of regularization is exploited. A compact quadratic optimization procedure is presented which obviates the requirement to invert large matrices, encountered in typical applications, and thereby reduce computation time and errors due to numerical round-off. The solution to the general linear problem Ax y with side conditions can be interpreted as a quadratic optimization problem. Such a requirement arises in the solution of illposed problems where the method of regularization is exploited. A compact quadratic optimization procedure is presented which obviates the requirement to invert large matrices, encountered in typical applications, and thereby reduce computation time and errors due to numerical round-off. P. H. Calamai and L. N. Vicente. ALGORITHM 728: Fortran subroutines for generating quadratic bilevel programming test problems. ACM Transactions on Mathematical Software, 20(1), 120–123, 1994a. P. H. Calamai and L. N. Vicente. Generating quadratic bilevel programming test problems. ACM Transactions on Mathematical Software, 20(1), 103–119, 1994b. P. H. Calamai, J. J. Júdice, and L. N. Vicente. Generation of disjointly constrained bilinear programming test problems. Computational Optimization and Applications, 1, 299–306, 1992. P. H. Calamai, L. N. Vicente, and J. J. Júdice. A new technique for generating quadraticprogramming test problems. Mathematical Programming, 61(2), 215–231, 1993. Abstract. This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated. Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems. This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated. Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems. A. J. Calise. Statistical design of sampled-data control systems via quadratic programming. Technical report, Univ Pennsylvania, PA, USA, 1968. Abstract. Quadratic programming is applied to the statistical design of Linear sampled-data control systems. Given a fixed plant, a compensator is chosen which minimizes the mean-square value of an error sequence subject to a set of constraints. The system inputs are described in terms of sampled values from autocorrelation and cross-correlation functions, eliminating the need for the analytical expressions required in the Wiener-Hopf equation. Constraints which characterize the closed-loop response to deterministic inputs and constraints which limit the complexity of the compensating net-work can be simultaneously employed. The solution generates the parameters of the optimum compensator. The case where the error signal is not sampled and where the mean-square value of the error is the index of the performance is also considered. Quadratic programming is applied to the statistical design of Linear sampled-data control systems. Given a fixed plant, a compensator is chosen which minimizes the mean-square value of an error sequence subject to a set of constraints. The system inputs are described in terms of sampled values from autocorrelation and cross-correlation functions, eliminating the need for the analytical expressions required in the Wiener-Hopf equation. Constraints which characterize the closed-loop response to deterministic inputs and constraints which limit the complexity of the compensating net-work can be simultaneously employed. The solution generates the parameters of the optimum compensator. The case where the error signal is not sampled and where the mean-square value of the error is the index of the performance is also considered. E. K. Can. A quadratic-programming solution to cost-time trade-off for CPM. In ‘Applied Simulation and Modelling—ASM ’85’, Vol. 3, pp. 253–256, 1985. W. Candler and R. J. Townsley. The maximization of a quadratic function of variables subject to linear inequalities. Management Science, 10, 515–523, 1964. 20 A QUADRATIC PROGRAMMING BIBLIOGRAPHY E. Canestrelli, S. Giove, and R. Fuller. Stability in possibilistic quadratic programming. Fuzzy Sets and Systems, 82(1), 51–56, 1996. Abstract. We show that possibilistic quadratic programs with crisp decision variables and continuous fuzzy number coefficients are well-posed, i.e. small changes in the membership function of the coefficients may cause only a small deviation in the possibility distribution of the objective function. We show that possibilistic quadratic programs with crisp decision variables and continuous fuzzy number coefficients are well-posed, i.e. small changes in the membership function of the coefficients may cause only a small deviation in the possibility distribution of the objective function. J. Canisius and J. L. van Hemmen. A polynomial time algorithm in general quadratic programming and ground-state properties of spin glasses. Europhysics Letters, 1(7), 319–326, 1986. Abstract. An algorithm is presented which finds a surprisingly good approximation to the global minimum of a not necessarily convex, quadratic function in N variables, restricted to a N-dimensional cube. For instance, all Ising Hamiltonians with pair interactions belong to this class. In general, the time complexity of the algorithm is O N4 . For nearest-neighbour interactions, it reduces to O N3 . The algorithm is used to find the ground state of various Ising spin glass models and to study the zero-temperature behavior of the magnetization as a function of the external field h in both and three dimensions. It is found that the two-dimensional +or-J model has a nonzero magnetization as h 0. An algorithm is presented which finds a surprisingly good approximation to the global minimum of a not necessarily convex, quadratic function in N variables, restricted to a N-dimensional cube. For instance, all Ising Hamiltonians with pair interactions belong to this class. In general, the time complexity of the algorithm is O N4 . For nearest-neighbour interactions, it reduces to O N3 . The algorithm is used to find the ground state of various Ising spin glass models and to study the zero-temperature behavior of the magnetization as a function of the external field h in both and three dimensions. It is found that the two-dimensional +or-J model has a nonzero magnetization as h 0. M. D. Canon and J. H. Eaton. A new algorithm for a class of quadratic programming problems with application to control. SIAM Journal on Control, 4, 34–45, 1996. J. M. Cao. Necessary and sufficient condition for local minima of a class of nonconvex quadratic programs. Mathematical Programming, 69(3), 403–411, 1995. M. Capurso. A quadratic programming approach to the impulsive loading analysis of rigid plastic structures. Meccanica, 7(1), 45–57, 1972. Abstract. This paper discusses the dynamic problem of rigid plastic structures subjected to impulsive loading. A couple of ’dual’ extremum theorems reduces the problem to the optimization of convex quadratic functions subject to linear equalities and equations: the first theorem takes as variables stress and accelerations, the second accelerations and plastic multiplier rates. The problem is discussed in matrix notation on the basis of finite element discretization of the structure and piecewise linear approximation of the yield surfaces, using some quadratic programming concepts. The procedure is illustrated by a simple numerical example. This paper discusses the dynamic problem of rigid plastic structures subjected to impulsive loading. A couple of ’dual’ extremum theorems reduces the problem to the optimization of convex quadratic functions subject to linear equalities and equations: the first theorem takes as variables stress and accelerations, the second accelerations and plastic multiplier rates. The problem is discussed in matrix notation on the basis of finite element discretization of the structure and piecewise linear approximation of the yield surfaces, using some quadratic programming concepts. The procedure is illustrated by a simple numerical example. R. J. Caron. Parametric quadratic programming. Windsor Mathematics Report 86-01, Department of Mathematics, University of Windsor, Ontario, Canada, 1986. T. J. Carpenter and D. F. Shanno. An interior point method for quadratic programs based on conjugate projected gradients. Computational Optimization and Applications, 2(1), 65– 28, 1993. Abstract. We propose an interior point method for large-scale convex quadratic programming where no assumptions are made about the sparsity structure of the quadratic coefficient matrix Q. The interior point method described is a doubly iterative algorithm that invokes a conjugate projected gradient procedure to obtain the search direction. The effect is that Q appears in a conjugate direction routine rather than in a matrix factorization. By doing this, the matrices to be factored have the same nonzero structure as those in linear programming. Further, one variant of this method is theoretically convergent with only one matrix factorization throughout the procedure. We propose an interior point method for large-scale convex quadratic programming where no assumptions are made about the sparsity structure of the quadratic coefficient matrix Q. The interior point method described is a doubly iterative algorithm that invokes a conjugate projected gradient procedure to obtain the search direction. The effect is that Q appears in a conjugate direction routine rather than in a matrix factorization. By doing this, the matrices to be factored have the same nonzero structure as those in linear programming. Further, one variant of this method is theoretically convergent with only one matrix factorization throughout the procedure. T. J. Carpenter, I. J. Lustig, J. M. Mulvey, and D. F. Shanno. Higher-order predictor-corrector interior point methods with application to quadratic objectives. SIAM Journal on Optimization, 3(4), 696–725, 1993a. N. I. M. GOULD & PH. L. TOINT 21 T. J. Carpenter, I. J. Lustig, J. M. Mulvey, and D. F. Shanno. Separable quadratic programming via a primal-dual interior point method and its use in a sequential procedure. ORSA Journal on Computing, 5(2), 182–191, 1993b. Abstract. Extends a primal-dual interior point procedure for linear programs to the case of convex separable quadratic objectives. Included are efficient procedures for: attaining primal and dual feasibility, variable upper bounding, and free variables. A sequential procedure that invokes the quadratic solver is proposed and implemented for solving linearly constrained convex separable nonlinear programs. Computational results are provided for several large test cases from stochastic programming. The proposed methods compare favorably with MINOS, especially for the larger examples. The nonlinear programs range in size up to 8700 constraints and 22000 variables. Extends a primal-dual interior point procedure for linear programs to the case of convex separable quadratic objectives. Included are efficient procedures for: attaining primal and dual feasibility, variable upper bounding, and free variables. A sequential procedure that invokes the quadratic solver is proposed and implemented for solving linearly constrained convex separable nonlinear programs. Computational results are provided for several large test cases from stochastic programming. The proposed methods compare favorably with MINOS, especially for the larger examples. The nonlinear programs range in size up to 8700 constraints and 22000 variables. J. L. Carpentier, G. Cotto, and P. L. Niederlander. New concepts for automatic generation control in electric power systems using parametric quadratic programming. In A. AlonsoConcheiro, ed., ‘Real Time Digital Control Applications. Proceedings of the IFAC/IFIP Symposium. Pergamon, Oxford, England’, pp. 595–600, 1984. Abstract. New concepts for automatic generation control in electric power systems are presented, where the two components of automatic generation control, load frequency control and economic dispatch are performed at the same rate, i.e. a few seconds, and where economic dispatch takes network security into account. This gives network security and good transients, avoiding contradictory actions of load frequency control and economic dispatch on the generating units. The corner stone of the solution is the use of a new fast on-line optimal power flow, using a new parametric quadratic programming method, which is presented in details. New concepts for automatic generation control in electric power systems are presented, where the two components of automatic generation control, load frequency control and economic dispatch are performed at the same rate, i.e. a few seconds, and where economic dispatch takes network security into account. This gives network security and good transients, avoiding contradictory actions of load frequency control and economic dispatch on the generating units. The corner stone of the solution is the use of a new fast on-line optimal power flow, using a new parametric quadratic programming method, which is presented in details. P. Carraresi, F. Farinaccio, and F. Malucelli. Testing optimality for quadratic 0-1 problems. Technical Report TR-95-11, Dipartimento di Informatica, Università di Pisa, Italy, 1995. Abstract. The issue tackled is testing whether a given solution of a quadratic 0-1 problem is optimal. The paper presents an algorithm based on the necessary and sufficient optimality condition introduced by HirriartUrruty for general convex problems. A measure of the quality of the solution is provided. Computational results show the applicability of the method. The method is extended to constrained quadratic 0-1 problems such as quadratic assignment and quadratic knapsack. The issue tackled is testing whether a given solution of a quadratic 0-1 problem is optimal. The paper presents an algorithm based on the necessary and sufficient optimality condition introduced by HirriartUrruty for general convex problems. A measure of the quality of the solution is provided. Computational results show the applicability of the method. The method is extended to constrained quadratic 0-1 problems such as quadratic assignment and quadratic knapsack. E. Casas and C. Pola. An algorithm for indefinite quadratic programming based on a partial Cholesky factorization. RAIRO-Recherche Operationnelle-Operations Research, 27(4), 401–426, 1993. Abstract. A new algorithm is described for quadratic programming that is based on a partial Cholesky factorization that uses a diagonal pivoting strategy and allows computation of the null of negative curvature directions. The algorithm is numerically stable and has shown efficiency in solving positive-definite and indefinite problems. It is specially interesting in indefinite cases because the initial point does not need to be a vertex of the feasible set. The authors thus avoid introducing artificial constraints in the problem, which turns out to be very efficient in parametric programming. At the same time, techniques for updating matrix factorizations are used. A new algorithm is described for quadratic programming that is based on a partial Cholesky factorization that uses a diagonal pivoting strategy and allows computation of the null of negative curvature directions. The algorithm is numerically stable and has shown efficiency in solving positive-definite and indefinite problems. It is specially interesting in indefinite cases because the initial point does not need to be a vertex of the feasible set. The authors thus avoid introducing artificial constraints in the problem, which turns out to be very efficient in parametric programming. At the same time, techniques for updating matrix factorizations are used. Y. Chabrillac and J.-P. Crouzeix. Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra and its Applications, 63, 283–292, 1984. T. F. Chan, J. A. Olkin, and D. W. Cooley. Solving quadratically constrained least squares using black box solvers. BIT, 32, 481–495, 1992. S. W. Chang. A method for quadratic programming. Naval Research Logistics Quarterly, 33(3), 479–487, 1986. 22 A QUADRATIC PROGRAMMING BIBLIOGRAPHY Abstract. A solution to the quadratic programming is presented with the constraint of the form Ax b using the linear complementary problem approach. A solution to the quadratic programming is presented with the constraint of the form Ax b using the linear complementary problem approach. Y. Y. Chang and R. W. Cottle. Least-index resolution of degeneracy in quadratic programming. Mathematical Programming, 18(2), 127–137, 1980. Abstract. Combines least-index pivot selection rules with Keller’s algorithm for quadratic programming to obtain a finite method for processing degenerate problems. Combines least-index pivot selection rules with Keller’s algorithm for quadratic programming to obtain a finite method for processing degenerate problems. A. Charnes and J. Semple. Practical error bounds for a class of quadratic programming problems. Informatica, 2(3), 352–366, 1991. Abstract. The absolute error between an approximate feasible solution, generated via a dual formulation, and the true optimal solution is measured. These error bounds involve considerably less computational work than existing estimates. The absolute error between an approximate feasible solution, generated via a dual formulation, and the true optimal solution is measured. These error bounds involve considerably less computational work than existing estimates. B. Chen and P. T. Harker. A noninterior continuation method for quadratic and linear programming. SIAM Journal on Optimization, 3(3), 503–515, 1993. M. Chen and J. A. Filar. Hamiltonian cycles, quadratic programming, and ranking of extreme points. In C. Floudas and P. Pardalos, eds, ‘Global Optimization’, pp. 32–9. Princeton University Press, USA, 1992. Y.-H. Chen and S. C. Fang. Neurocomputing with time delay analysis for solving convex quadratic programming problems. IEEE Transactions on Neural Networks, p. (to appear), 1999. Y. H. Chen and S. C. Fang. Neurocomputing with time delay analysis for solving convex quadratic programming problems. IEEE Transactions on Neural Networks, 11(1), 230– 240, 2000. Abstract. This paper presents a neural-network computational scheme with time-delay consideration for solving convex quadratic programming problems. Based on some known results, a delay margin is explicitly determined for the stability of the neural dynamics, under which the states of the neural network does not oscillate. The configuration of the proposed neural network is provided. Operational characteristics of the neural network are demonstrated via numerical examples. This paper presents a neural-network computational scheme with time-delay consideration for solving convex quadratic programming problems. Based on some known results, a delay margin is explicitly determined for the stability of the neural dynamics, under which the states of the neural network does not oscillate. The configuration of the proposed neural network is provided. Operational characteristics of the neural network are demonstrated via numerical examples. Z. Chen and N. Y. Deng. Some algorithms for the convex quadratic programming problem via the ABS approach. Optimization Methods and Software, 8(2), 157–170, 1997. F. T. Cheng, T. H. Chen, and Y. Y. Sun. Efficient algorithm for resolving manipulator redundancy—the compact QP method. In ‘1992 IEEE International Conference on Robotics and Automation : Proceedings’, Vol. 1–3, pp. 508–513, 1992a. F. T. Cheng, T. H. Chen, Y. S. Wang, and Y. Y. Sun. Efficient algorithm for resolving manipulator redundancy-the compact QP method. In ‘Proceedings. 1992 IEEE International Conference on Robotics And Automation. IEEE Comput. Soc. Press, Los Alamitos, CA, USA’, Vol. 1, pp. 508–513, 1992b. Abstract. Due to hardware limitations, physical constraints, such as joint rate bounds and joint angle limits, always exist. In the present work, these constraints are included in the general formulation of the redundant inverse kinematic problem. To take into account these physical constraints, the computationally efficient compact QP (quadratic programming) method is derived to resolve the kinematic redundancy problem. In addition, the compact-inverse QP method is developed to remedy the singularity problem. The compact QP Due to hardware limitations, physical constraints, such as joint rate bounds and joint angle limits, always exist. In the present work, these constraints are included in the general formulation of the redundant inverse kinematic problem. To take into account these physical constraints, the computationally efficient compact QP (quadratic programming) method is derived to resolve the kinematic redundancy problem. In addition, the compact-inverse QP method is developed to remedy the singularity problem. The compact QP N. I. M. GOULD & PH. L. TOINT 23 (compact and inverse QP) method makes use of the compact formulation to obtain the general solutions and to eliminate the equality constraints. As such, the variables are decomposed into basic and free variables, and the basic variables are expressed by the free variables. Thus, the problem size is reduced and it only requires an optimization algorithm, such as QP, for the free variables subject to pure inequality constraints. This approach will expedite the optimization process and make real-time implementation possible. F. T. Cheng, T. H. Chen, Y. S. Wang, and Y. Y. Sun. Obstacle avoidance for redundant manipulators using the compact QP method. In ‘Proceedings IEEE International Conference on Robotics and Automation. IEEE Comput. Soc. Press, Los Alamitos, CA, USA’, Vol. 3, pp. 262–269, 1993. Abstract. The compact QP (quadratic programming) method is proposed to resolve the obstacle avoidance problem for a redundant manipulator. The drift-free criterion is considered when a redundant manipulator performs a repeated motion. Due to the computational efficiency and versatility of the compact QP method, real-time implementation is able to be achieved, and physical limitations such as joint rate bounds and joint angle limits can be easily taken into account. An example is given to demonstrate that this method is able to avoid the throat of a cavity, and to remedy the drift problem while a primary goal of the manipulators is carried out. Simulation results show that multiple goals can easily be fulfilled by this method. The compact QP (quadratic programming) method is proposed to resolve the obstacle avoidance problem for a redundant manipulator. The drift-free criterion is considered when a redundant manipulator performs a repeated motion. Due to the computational efficiency and versatility of the compact QP method, real-time implementation is able to be achieved, and physical limitations such as joint rate bounds and joint angle limits can be easily taken into account. An example is given to demonstrate that this method is able to avoid the throat of a cavity, and to remedy the drift problem while a primary goal of the manipulators is carried out. Simulation results show that multiple goals can easily be fulfilled by this method. F. T. Cheng, R. J. Sheu, T. H. Chen, Y. S. Wang, and F. C. Kung. The improved compact QP method for resolving manipulator redundancy. In ‘IROS ’94. Proceedings of the IEEE/RSJ/GI International Conference on Intelligent Robots and Systems. Advanced Robotic Systems and the Real World. IEEE, New York, NY, USA’, Vol. 2, pp. 1368–1375, 1994. Abstract. The compact QP method is an effective and efficient algorithm for resolving the manipulator redundancy under inequality constraints. In this paper, a more computationally efficient scheme which will improve the efficiency of the compact QP method-the improved compact QP method -is developed. With the technique of workspace decomposition, the redundant inverse kinematics problem can be decomposed into two subproblems. Thus, the size of the redundancy problem can be reduced. For an n degree-of-freedom spatial redundant manipulator, instead of a 6n matrix, only a 3 n 3 matrix is needed to be manipulated by Gaussian elimination with partial pivoting for selecting the free variables. The simulation results on the CESAR manipulator indicate that the speedup of the compact QP method as compared with the original QP method is about 4.3. Furthermore, the speedup of the improved compact QP method is about 5.6. Therefore, it is believed that the improved compact QP method is one of the most efficient and effective optimization algorithm for resolving the manipulator redundancy under inequality constraints. The compact QP method is an effective and efficient algorithm for resolving the manipulator redundancy under inequality constraints. In this paper, a more computationally efficient scheme which will improve the efficiency of the compact QP method-the improved compact QP method -is developed. With the technique of workspace decomposition, the redundant inverse kinematics problem can be decomposed into two subproblems. Thus, the size of the redundancy problem can be reduced. For an n degree-of-freedom spatial redundant manipulator, instead of a 6n matrix, only a 3 n 3 matrix is needed to be manipulated by Gaussian elimination with partial pivoting for selecting the free variables. The simulation results on the CESAR manipulator indicate that the speedup of the compact QP method as compared with the original QP method is about 4.3. Furthermore, the speedup of the improved compact QP method is about 5.6. Therefore, it is believed that the improved compact QP method is one of the most efficient and effective optimization algorithm for resolving the manipulator redundancy under inequality constraints. C. C. N. Chu and D. F. Wong. A quadratic programming approach to simultaneous buffer insertion/sizing and wire sizing. IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, 18(6), 787–798, 1999. Abstract. In this paper, we present a completely new approach to the problem of delay minimization by simultaneous buffer insertion and wire sizing for a wire. We show that the problem can be formulated as a convex quadratic program, which is known to be solvable in polynomial time. Nevertheless, we explore some special properties of our problem and derive an optimal and very efficient algorithm, modified active set method (MASM), to solve the resulting program. Given m buffers and a set of m discrete choices of wire width, the running time of our algorithm is O mn2 and is independent of the wire length in practice. For example, an instance of 100 buffers and 100 choices of wire width can be solved in 0.92 s. In addition, we extend MASM to consider simultaneous buffer insertion, buffer sizing, and wire sizing. The resulting algorithm MASM-BS is again optimal and very efficient. For example, with six choices of buffer size and 10 choices of wire width, the optimal solution for a 15000 μ m long wire can be found in 0.05 s. Besides, our formulation is so versatile that it is easy to consider other objectives like wire area or power dissipation, or to add constraints to the solution. Also, wire capacitance lookup tables, or very general wire capacitance models which can capture area capacitance, fringing capacitance, coupling capacitance, etc. can be used. In this paper, we present a completely new approach to the problem of delay minimization by simultaneous buffer insertion and wire sizing for a wire. We show that the problem can be formulated as a convex quadratic program, which is known to be solvable in polynomial time. Nevertheless, we explore some special properties of our problem and derive an optimal and very efficient algorithm, modified active set method (MASM), to solve the resulting program. Given m buffers and a set of m discrete choices of wire width, the running time of our algorithm is O mn2 and is independent of the wire length in practice. For example, an instance of 100 buffers and 100 choices of wire width can be solved in 0.92 s. In addition, we extend MASM to consider simultaneous buffer insertion, buffer sizing, and wire sizing. The resulting algorithm MASM-BS is again optimal and very efficient. For example, with six choices of buffer size and 10 choices of wire width, the optimal solution for a 15000 μ m long wire can be found in 0.05 s. Besides, our formulation is so versatile that it is easy to consider other objectives like wire area or power dissipation, or to add constraints to the solution. Also, wire capacitance lookup tables, or very general wire capacitance models which can capture area capacitance, fringing capacitance, coupling capacitance, etc. can be used. 24 A QUADRATIC PROGRAMMING BIBLIOGRAPHY C. S. Chung and D. Gale. A complementarity algorithm for optimal stationary programs in growth models with quadratic utility. Technical Report ORC 81-10, Operations Research Center, University of California, Berkeley, CA, USA, 1981. S. J. Chung and K. G. Murty. Polynomially bounded ellipsoid algorithm for convex quadratic programming. In O. L. Mangasarian, R. R. Meyer and S. M. Robinson, eds, ‘Nonlinear Programming, 4’, pp. 439–485, Academic Press, London and New York, 1981a. S. J. Chung and K. G. Murty. Polynomially bounded ellipsoid algorithms for convex quadratic programming. Methods of Operations Research, 40, 63–66, 1981b. Abstract. Let B, b be respectively a given square nonsingular integer matrix of order n, and an integer column vector in IRn. It is required to find the nearest point to b in the Cone Pos B x : x Bz z 0 . This leads to the linear complementarity problem: find w w1 wn , z z1 zn satisfying W BT B z BT b, w 0 z 0, wz 0. T. T. Chung. Analysis of plate bending by the quadratic programming approach. Technical report, Washington Univesity, St. Louis, MO, USA, 1974. Let B, b be respectively a given square nonsingular integer matrix of order n, and an integer column vector in IRn. It is required to find the nearest point to b in the Cone Pos B x : x Bz z 0 . This leads to the linear complementarity problem: find w w1 wn , z z1 zn satisfying W BT B z BT b, w 0 z 0, wz 0. T. T. Chung. Analysis of plate bending by the quadratic programming approach. Technical report, Washington Univesity, St. Louis, MO, USA, 1974. Abstract. Finite element analysis of plate bending is interpreted as a quadratic programming problem. The total potential energy, expressed in terms of the coefficients of the approximating polynomials is the objective function the minimum of which is sought subject to linear equality constraints. The constraints require satisfaction of all kinematic boundary conditions and inter-element continuity conditions. Convergence characteristics of this approach with respect to increasing orders of polynomial approximation, as well as with respect to progressively reduced element sizes, are discussed and illustrated with a number of examples. Advantages of the proposed approach are discussed, and topics requiring further investigations are outlined. Finite element analysis of plate bending is interpreted as a quadratic programming problem. The total potential energy, expressed in terms of the coefficients of the approximating polynomials is the objective function the minimum of which is sought subject to linear equality constraints. The constraints require satisfaction of all kinematic boundary conditions and inter-element continuity conditions. Convergence characteristics of this approach with respect to increasing orders of polynomial approximation, as well as with respect to progressively reduced element sizes, are discussed and illustrated with a number of examples. Advantages of the proposed approach are discussed, and topics requiring further investigations are outlined. L. Churilov, D. Ralph, and M. Sniedovich. A note on composite concave quadratic programming. Operations Research Letters, 23(3–5), 163–169, 1998. Abstract. We present a pivotal-based algorithm for the global minimization of composite concave quadratic functions subject to linear constraints. It is shown that certain subclasses of this family yield easy-to-solve line search subproblems. Since the proposed algorithm is equivalent in efficiency to a standard parametric complementary pivoting procedure, the implication is that conventional parametric quadratic programming algorithms can now be used as tools for the solution of much wider class of complex global optimization problems. We present a pivotal-based algorithm for the global minimization of composite concave quadratic functions subject to linear constraints. It is shown that certain subclasses of this family yield easy-to-solve line search subproblems. Since the proposed algorithm is equivalent in efficiency to a standard parametric complementary pivoting procedure, the implication is that conventional parametric quadratic programming algorithms can now be used as tools for the solution of much wider class of complex global optimization problems. T. F. Coleman and L. A. Hulbert. A direct active set algorithm for large sparse quadratic programs with simple bounds. Mathematical Programming, Series B, 45(3), 373–406, 1989. T. F. Coleman and L. A. Hulbert. A globally and superlinearly convergent algorithm for convex quadratic programs with simple bounds. SIAM Journal on Optimization, 3(2), 298–321, 1993. T. F. Coleman and Y. Li. A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables. SIAM Journal on Optimization, 6(4), 1040– 1058, 1996. Abstract. We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. The method applies to a general (indefinite) quadratic function for which a local minimum subject to bounds is required and is particularly suitable for the large-scale problem. Our new method exhibits strong convergence properties and global and second-order convergence and appears to have significant practical potential. Strictly feasible We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. The method applies to a general (indefinite) quadratic function for which a local minimum subject to bounds is required and is particularly suitable for the large-scale problem. Our new method exhibits strong convergence properties and global and second-order convergence and appears to have significant practical potential. Strictly feasible N. I. M. GOULD & PH. L. TOINT 25 points are generated. We provide experimental results on moderately large and sparse problems based on both sparse Cholesky and preconditioned conjugate gradient linear solvers. T. F. Coleman and J. G. Liu. An interior Newton method for quadratic programming. Mathematical Programming, 85(3), 491–523, 1999. Abstract. We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential. We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential. T. F. Coleman and J. G. Liu. An exterior Newton method for strictly convex quadratic programming. Computational Optimization and Applications, 15(1), 5–32, 2000. Abstract. We propose an exterior Newton method for strictly convex quadratic programming (QP) problems. This method is based on a dual formulation: a sequence of points is generated which monotonically decreases the dual objective function. We show that the generated sequence converges globally and quadratically to the solution (if the QP is feasible and certain nondegeneracy assumptions are satisfied). Measures for detecting infeasibility are provided. The major computation in each iteration is to solve a KKT-like system. Therefore, given an effective symmetric sparse linear solver, the proposed method is suitable for large sparse problems. Preliminary numerical results are reported. We propose an exterior Newton method for strictly convex quadratic programming (QP) problems. This method is based on a dual formulation: a sequence of points is generated which monotonically decreases the dual objective function. We show that the generated sequence converges globally and quadratically to the solution (if the QP is feasible and certain nondegeneracy assumptions are satisfied). Measures for detecting infeasibility are provided. The major computation in each iteration is to solve a KKT-like system. Therefore, given an effective symmetric sparse linear solver, the proposed method is suitable for large sparse problems. Preliminary numerical results are reported. D. C. Collins. Terminal state dynamic programming: quadratic costs, linear differential equations. Journal of Mathematical Analysis and Applications, 31(2), 235–253, 1970. Abstract. A fairly general class of control problems can be posed in terms of minimizing a cost functional involving the state of the system to be controlled and the control exerted over a fixed interval of time. The state and control variables are related by a state equation, often with additional constraints of various forms upon the control or state. That is, it is desired to find miny tεY p x T ! A fairly general class of control problems can be posed in terms of minimizing a cost functional involving the state of the system to be controlled and the control exerted over a fixed interval of time. The state and control variables are related by a state equation, often with additional constraints of various forms upon the control or state. That is, it is desired to find miny tεY p x T !