Conic approximation to quadratic optimization with linear complementarity constraints
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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.
KeywordsCone of nonnegative quadratic functions Conic approximation Linear complementarity constraints
Mathematics Subject Classification90C26 90C34
This work has been partially supported by the National Natural Science Foundation of China under Grant Numbers 11171177, 11371216, 11526186 and 11571029, the Zhejiang Provincial Natural Science Foundation of China under Grant Number LQ16A010010, and US Army Research Office Grant Number W911NF-15-1-0223. The authors would also like to thank the editor and reviewers for their most valuable comments and suggestions.
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