Conic approximation to quadratic optimization with linear complementarity constraints

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• 作者：Jing Zhou ; Shu-Cherng Fang ; Wenxun Xing
• 关键词：Cone of nonnegative quadratic functions ; Conic approximation ; Linear complementarity constraints
• 刊名：Computational Optimization and Applications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：66
• 期：1
• 页码：97-122
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Optimization; Operations Research, Management Science; Operation Research/Decision Theory; Statistics, general; Convex and Discrete Geometry;
• 出版者：Springer US
• ISSN：1573-2894
• 卷排序：66

文摘

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We 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 \(\epsilon \)-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.