Conic Reformulations and Approximations to Some Subclasses of Nonconvex
نویسندگان
چکیده
DENG, ZHIBIN. Conic Reformulations and Approximations to Some Subclasses of Nonconvex Quadratic Programming Problems. (Under the direction of Dr. Shu-Cherng Fang.) In this dissertation, some subclasses of nonconvex quadratic programming problems are studied. We first study the nonconvex quadratic programming problem over the standard simplex with application to copositive matrix detection. A sequence of linear conic approximations are derived to bound the original problem by using semidefinite programming techniques. An algorithm based on the adaptive approach is developed to detect the copositivity of a given matrix. Then, we study the nonconvex quadratic programming problem over a set of convex quadratic constraints. A conic reformulation and approximation with an adaptive scheme for this problem is also developed. We proved that an -optimal solution can be obtained in a finite number of iterations using the proposed algorithm. Finally, we extend our study to the bounded nonconvex quadratically constrained quadratic programming problem. A branch-andcut algorithm is developed to solve this problem based on some generalized linear and quadratic polar cuts. The finite termination of the proposed algorithm is proved and our numerical results confirm its superior performance over other known algorithms in the literature. Directions for future research are included at the end. Conic Reformulations and Approximations to Some Subclasses of Nonconvex Quadratic Programming Problems
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