Computational Optimization and Applications

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

On convex quadratic programs with linear complementarity constraints



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.


  1. 1.
    Bienstock, D.: Eigenvalue techniques for proving bounds for convex objective, nonconvex programs. In: IPCO, vol. 29 (2010) Google Scholar
  2. 2.
    Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Eaves, B.C.: On quadratic programing. Manag. Sci. 17, 698–711 (1971) MATHCrossRefGoogle Scholar
  4. 4.
    Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Electrical Engineering Department, Stanford University (2002) Google Scholar
  5. 5.
    Fazel, M., Hindi, H., Boyd, S.: Rank minimization and applications in system theory. In: Proceedings of the 2004 American Control Conference, Boston, Massachusetts, 30–July 2, 2004, pp. 3273–3278 (2004) Google Scholar
  6. 6.
    Han, Z.: A SAT solver implemented in MATLAB Accessed September 2011
  7. 7.
    Hooker, J.N.: Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction. Wiley, New York (2000) MATHCrossRefGoogle Scholar
  8. 8.
    Hooker, J.N.: Integrated Methods for Optimization. Springer, New York (2006) Google Scholar
  9. 9.
    Hooker, J.N., Ottosson, G.: Logic-based benders decomposition. Math. Program. 96, 33–60 (2003) MathSciNetMATHGoogle Scholar
  10. 10.
    Hu, J.: On linear programs with linear complementarity constraints. Ph.D. thesis, Mathematical Sciences, Rensselaer Polytechnic Institute (2009) Google Scholar
  11. 11.
    Hu, J., Mitchell, J.E., Pang, J.S., Bennett, K.P., Kunapuli, G.: On the global resolution of linear programs with linear complementarity constraints. SIAM J. Optim. 19, 445–471 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hu, J., Mitchell, J.E., Pang, J.S.: An LPCC approach to nonconvex quadratic programs. Math. Program. 133, 243–277 (2012) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hu, J., Mitchell, J.E., Pang, J.S., Yu, B.: On linear programs with linear complementarity constraints. J. Glob. Optim. 53, 29–51 (2012) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jiang, H., Ralph, D.: QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints. Comput. Optim. Appl. 13, 25–59 (1999) MathSciNetCrossRefGoogle Scholar
  15. 15.
    KNITRO User’s Manual Version 6.0. Accessed February 2012
  16. 16.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996) CrossRefGoogle Scholar
  17. 17.
    MacMPEC Test Problems. Accessed September 2011
  18. 18.
    Mitchell, J.E., Pang, J.S., Yu, B.: Obtaining tighter relaxations of linear programs with complementarity constraints. In: Modeling and Optimization: Theory and Applications. Springer Proceedings in Mathematics and Statistics, vol. 21 (2012) Google Scholar
  19. 19.
    Pang, J.S.: Three modeling paradigms in mathematical programming. Math. Program. 125, 297–323 (2010) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3—a Matlab software package for semidefinite programming. Optim. Methods Softw. 11, 545–581 (1999) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Yu, B.: A branch and cut approach to linear programs with linear complementarity constraints. Ph.D. thesis, Mathematical Sciences, Rensselaer Polytechnic Institute (2011) Google Scholar

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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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