Noname manuscript No. (will be inserted by the editor) On Convex Quadratic Programs with Linear Complementarity Constraints Lijie Bai · John E.Mitchell · Jong-Shi Pang Received: date / Accepted: date 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. Keywords Convex quadratic programming · Logical Benders decomposition · Satisfiability constraints · Semidefinite programming. 1 Introduction Mathematical programs with complementarity constraints (MPCCs) are constrained optimization problems subject to certain distinguished disjunctive constraints expressed by the complementarity relation between pairs of nonnegative variables. The origin of MPCCs can be traced back to the class of Stackelberg games in economics. Extensive applications of MPCCs can be found in hierarchical (particularly bi-level) decision making, inverse optimization, parameter identification, optimal design, and many other contexts; various examples of MPCCs are documented in [16,22]. Linear and convex quadratic programs with complementarity constraints (LPCCs/QPCCs, The work of Mitchell was supported by the National Science Foundation under grant DMS-0715446 and by the Air Force Office of Sponsored Research under grant FA9550-08-1-0081 and FA9550-11-1-0260. The work of Pang was supported by the U.S.A. National Science Foundation grant CMMI 0969600 and by the Air Force Office of Sponsored Research under grants FA9550-08-1-0061 and FA9550-11-1-0151. Lijie Bai Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA E-mail: bail@rpi.edu John E.Mitchell Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA E-mail: mitchj@rpi.edu Jong-Shi Pang Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA E-mail: jspang@illinois.edu