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Conic approximation to quadratic optimization with linear complementarity constraints

Author

Listed:
  • Jing Zhou

    (Zhejiang University of Technology)

  • Shu-Cherng Fang

    (North Carolina State University)

  • Wenxun Xing

    (Tsinghua University)

Abstract

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 $$\epsilon $$ ϵ -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.

Suggested Citation

  • Jing Zhou & Shu-Cherng Fang & Wenxun Xing, 2017. "Conic approximation to quadratic optimization with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 66(1), pages 97-122, January.
    • Handle: RePEc:spr:coopap:v:66:y:2017:i:1:d:10.1007_s10589-016-9855-8
    DOI: 10.1007/s10589-016-9855-8
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    References listed on IDEAS

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    1. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    2. Joaquim Júdice & Ana Faustino & Isabel Ribeiro, 2002. "On the solution of NP-hard linear complementarity problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(1), pages 125-145, June.
    3. Lijie Bai & John Mitchell & Jong-Shi Pang, 2013. "On convex quadratic programs with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 54(3), pages 517-554, April.
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    Cited by:

    1. Shaoze Li & Zhibin Deng & Cheng Lu & Junhao Wu & Jinyu Dai & Qiao Wang, 2023. "An efficient global algorithm for indefinite separable quadratic knapsack problems with box constraints," Computational Optimization and Applications, Springer, vol. 86(1), pages 241-273, September.
    2. Immanuel M. Bomze & Bo Peng, 2023. "Conic formulation of QPCCs applied to truly sparse QPs," Computational Optimization and Applications, Springer, vol. 84(3), pages 703-735, April.

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