AFRL-OSR-VA-TR-2014-0126 Global Resolution of Convex Programs with Complementarity Constraints

Quadratic Convex Reformulation (QCR) is a technique that has been proposed for binary and mixed integer quadratic programs. In this paper, we extend the QCR method to convex quadratic programs with linear complementarity constraints (QPCCs). Due to the complementarity relationship between the nonnegative variables y and w, a term yDw can be added to the QPCC objective function, where D is a nonnegative diagonal matrix chosen to maintain the convexity of the objective function and the global resolution of the QPCC. Following the QCR method, the products of linear equality constraints can also be used to perturb the QPCC objective function, with the goal that the new QP relaxation provides a tighter lower bound. By solving a semidefinite program, an equivalent QPCC can be obtained whose QP relaxation is as tight as possible. In addition, we extend the QCR to a general quadratically constrained quadratic program (QCQP), of which the QPCC is a special example. Computational tests on QPCCs are presented. • J. Hu, J.E. Mitchell, and J.S. Pang. An LPCC approach to nonconvex quadratic programs. Mathematical Programming, Series A 133 (2012) 243–277. Abstract: Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving two linear programs with linear complementarity constraints, i.e., LPCCs. Specifically, this task can be divided into two LPCCs: the first confirms whether the QP is bounded below on the feasible set and, if not, computes a feasible ray on which the QP is unbounded; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can