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On convex quadratic programs with linear complementarity constraints

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  • Lijie Bai
  • John Mitchell
  • Jong-Shi Pang

Abstract

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the primal-dual relationship of a pair of convex quadratic programs and on a logical Benders scheme, an extreme ray/point generation procedure is developed, which relies on valid satisfiability constraints for the integer program. To improve this scheme, we propose a two-stage approach wherein the first stage solves the mixed integer quadratic program with pre-set upper bounds on the complementarity variables, and the second stage solves the program outside this bounded region by the Benders scheme. We report computational results with our method. We also investigate the addition of a penalty term y T Dw to the objective function, where y and w are the complementary variables and D is a nonnegative diagonal matrix. The matrix D can be chosen effectively by solving a semidefinite program, ensuring that the objective function remains convex. The addition of the penalty term can often reduce the overall runtime by at least 50 %. We report preliminary computational testing on a QP relaxation method which can be used to obtain better lower bounds from infeasible points; this method could be incorporated into a branching scheme. By combining the penalty method and the QP relaxation method, more than 90 % of the gap can be closed for some QPCC problems. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • 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.
  • Handle: RePEc:spr:coopap:v:54:y:2013:i:3:p:517-554
    DOI: 10.1007/s10589-012-9497-4
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    References listed on IDEAS

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    1. Jing Hu & John Mitchell & Jong-Shi Pang & Bin Yu, 2012. "On linear programs with linear complementarity constraints," Journal of Global Optimization, Springer, vol. 53(1), pages 29-51, May.
    2. B. Curtis Eaves, 1971. "On Quadratic Programming," Management Science, INFORMS, vol. 17(11), pages 698-711, July.
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    Cited by:

    1. Jong-Shi Pang & Meisam Razaviyayn & Alberth Alvarado, 2017. "Computing B-Stationary Points of Nonsmooth DC Programs," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 95-118, January.
    2. Jacquet, Quentin & van Ackooij, Wim & Alasseur, Clémence & Gaubert, Stéphane, 2024. "Quadratic regularization of bilevel pricing problems and application to electricity retail markets," European Journal of Operational Research, Elsevier, vol. 313(3), pages 841-857.
    3. Bai, Yun & Ouyang, Yanfeng & Pang, Jong-Shi, 2016. "Enhanced models and improved solution for competitive biofuel supply chain design under land use constraints," European Journal of Operational Research, Elsevier, vol. 249(1), pages 281-297.
    4. Jong-Shi Pang & Che-Lin Su & Yu-Ching Lee, 2015. "A Constructive Approach to Estimating Pure Characteristics Demand Models with Pricing," Operations Research, INFORMS, vol. 63(3), pages 639-659, June.
    5. Francisco Jara-Moroni & John E. Mitchell & Jong-Shi Pang & Andreas Wächter, 2020. "An enhanced logical benders approach for linear programs with complementarity constraints," Journal of Global Optimization, Springer, vol. 77(4), pages 687-714, August.
    6. Yu-Ching Lee & Jong-Shi Pang & John Mitchell, 2015. "An algorithm for global solution to bi-parametric linear complementarity constrained linear programs," Journal of Global Optimization, Springer, vol. 62(2), pages 263-297, June.
    7. 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.

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