On convex quadratic programs with linear complementarity constraints.

*(English)*Zbl 1295.90035Summary: 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 \(y^TDw\) 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.

##### MSC:

90C20 | Quadratic programming |

90C25 | Convex programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C22 | Semidefinite programming |

##### Keywords:

convex quadratic programming; logical Benders decomposition; satisfiability constraints; semidefinite programming
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\textit{L. Bai} et al., Comput. Optim. Appl. 54, No. 3, 517--554 (2013; Zbl 1295.90035)

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##### References:

[1] | Bienstock, D.: Eigenvalue techniques for proving bounds for convex objective, nonconvex programs. In: IPCO, vol. 29 (2010) · Zbl 1285.90038 |

[2] | CandĂ¨s, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009) · Zbl 1219.90124 |

[3] | Eaves, B.C.: On quadratic programing. Manag. Sci. 17, 698–711 (1971) · Zbl 0242.90040 |

[4] | Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Electrical Engineering Department, Stanford University (2002) |

[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) |

[6] | Han, Z.: A SAT solver implemented in MATLAB http://www.mathworks.com/matlabcentral/fileexchange/22284-satisfiability-solver Accessed September 2011 |

[7] | Hooker, J.N.: Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction. Wiley, New York (2000) · Zbl 0974.90001 |

[8] | Hooker, J.N.: Integrated Methods for Optimization. Springer, New York (2006) · Zbl 1103.68811 |

[9] | Hooker, J.N., Ottosson, G.: Logic-based benders decomposition. Math. Program. 96, 33–60 (2003) · Zbl 1023.90082 |

[10] | Hu, J.: On linear programs with linear complementarity constraints. Ph.D. thesis, Mathematical Sciences, Rensselaer Polytechnic Institute (2009) |

[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) · Zbl 1163.90031 |

[12] | Hu, J., Mitchell, J.E., Pang, J.S.: An LPCC approach to nonconvex quadratic programs. Math. Program. 133, 243–277 (2012) · Zbl 1244.90177 |

[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) · Zbl 1254.90111 |

[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) · Zbl 1040.90500 |

[15] | KNITRO User’s Manual Version 6.0. http://www.ziena.com/docs/Knitro60_UserManual.pdf . Accessed February 2012 |

[16] | Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996) · Zbl 0870.90092 |

[17] | MacMPEC Test Problems. http://wiki.mcs.anl.gov/leyffer/index.php/MacMPEC . Accessed September 2011 |

[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) · Zbl 1346.90686 |

[19] | Pang, J.S.: Three modeling paradigms in mathematical programming. Math. Program. 125, 297–323 (2010) · Zbl 1202.90254 |

[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) · Zbl 0997.90060 |

[21] | Yu, B.: A branch and cut approach to linear programs with linear complementarity constraints. Ph.D. thesis, Mathematical Sciences, Rensselaer Polytechnic Institute (2011) |

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