Computational Optimization and Applications

, Volume 54, Issue 3, pp 517–554 | Cite as

On convex quadratic programs with linear complementarity constraints

  • Lijie Bai
  • John E. Mitchell
  • Jong-Shi Pang


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.


Convex quadratic programming Logical Benders decomposition Satisfiability constraints Semidefinite programming 



Our thanks to two anonymous referees for their helpful and constructive comments.


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA
  2. 2.Department of Industrial and Enterprise Systems EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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