It is co-NP-complete to decide whether a given matrix is copositive or not. In this paper, this decision problem is transformed into a quadratic programming problem, which can be approximated by solving a sequence of ...
详细信息
It is co-NP-complete to decide whether a given matrix is copositive or not. In this paper, this decision problem is transformed into a quadratic programming problem, which can be approximated by solving a sequence of linear conic programming problems defined on the dual cone of the cone of nonnegative quadratic functions over the union of a collection of ellipsoids. Using linear matrix inequalities (LMI) representations, each corresponding problem in the sequence can be solved via semidefinite programming. In order to speed up the convergence of the approximation sequence and to relieve the computational effort of solving linear conic programming problems, an adaptive approximation scheme is adopted to refine the union of ellipsoids. The lower and upper bounds of the transformed quadratic programming problem are used to determine the copositivity of the given matrix. (C) 2013 Elsevier B.V. All rights reserved.
In this paper, a conic reformulation and approximation is proposed for solving a nonconvex quadratic programming problem subject to several convex quadratic constraints. The original problem is transformed into a line...
详细信息
In this paper, a conic reformulation and approximation is proposed for solving a nonconvex quadratic programming problem subject to several convex quadratic constraints. The original problem is transformed into a linear conic programming problem, which can be approximated by a sequence of linear conic programming problems over the dual cone of the cone of nonnegative quadratic functions. Since the dual cone of the cone of nonnegative quadratic functions has a linear matrix inequality representation, each linear conic programming problem in the sequence can be solved efficiently using the semidefinite programming techniques. In order to speed up the convergence of the approximation sequence and relieve the computational effort in solving the linear conic programming problems, an adaptive scheme is adopted in the proposed algorithm. We prove that the lower bounds generated by the linear conic programming problems converge to the optimal value of the original problem. Several numerical examples are used to illustrate how the algorithm works and the computational results demonstrate the efficiency of the proposed algorithm.
This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity *** provide a conic reformulation and its dual for the original problem such that these three...
详细信息
This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity *** provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an -optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.
This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality *** the feasible region is bounded,we show that an optimal solution of the problem can be found in poly...
详细信息
This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality *** the feasible region is bounded,we show that an optimal solution of the problem can be found in polynomial *** the feasible region is unbounded,a semidefinite programming(SDP)reformulation is constructed to find the optimal objective value of the original problem in polynomial *** addition,we provide two sufficient conditions,under which,if the optimal objective value is finite,we show the optimal solution of SDP reformulation can be decomposed into the original space to generate an optimal solution of the original problem in polynomial ***,a recession direction can be identified in polynomial *** examples are included to illustrate the effectiveness of the proposed approach.
暂无评论