Primal-dual Power Series Algorithm

23] C.L. Monma and A.J. Morton. Computational experimental with a dual aane variant of Karmarkar's method for linear programming. extension of Karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence. Techni-28] J. Renegar. A polynomial-time algorithm based on Newton's method for linear programming. Implementing an interior point method in a mathematical programming system. 28 REFERENCES 6] D. Bayer and J.C. Lagarias. The nonlinear geometry of linear programming I. AAne and projective scaling trajectories. 7] D. Bayer and J.C. Lagarias. The nonlinear geometry of linear programming II. Leg-endre transform coordinates and central trajectories. REFERENCES 27 O(n 3 + n 2 r) arithmetic operations, increases with r. When r = O(n) we still obtain O(n 3) arithmetic operations per iteration, which is the work per iteration required by all interior point based algorithms if no rank-one update trick is used 15]. The main purpose of this paper was to present a theoretical result. However, based on the good performance of both the primal aane 31] and dual aane scaling algorithms 1] 20] 23], we feel that the primal-dual aane scaling algorithm has the potential of becoming a competitive algorithm. For a practical implementation some modiications are required, such as: (1) introducing a larger step size computed by means of a ratio test in the rst order approximation algorithm or by means of a binary search in the higher order approximation algorithms; (2) determining an appropriate starting artiicial problem that gives a good initial starting point; and (3) making a good choice of r. Note that when r = 1, the primal-dual aane scaling algorithm described in section 3 can be viewed as a simultaneous application of an aane scaling algorithm to the primal and dual problems, which implies that both the primal and dual objective functions monoton-ically approach the optimal value. For a practical implementation, this suggests that two ratio tests performed independently in the primal and the dual spaces respectively, might outperform one ratio test done simultaneously in the primal-dual space, since a larger decrease in the duality gap would be obtained. On the other hand, the last strategy would be more conservative in the sense that it would keep the iterates from coming too close to the boundary of the primal-dual feasible region. Data structures and programming techniques for the implementation of Karmarkar's algorithm.