In this paper, we consider problem (P ) of minimizing a quadratic function q(x) =x t Qx + c t x of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal entries of Q by a vector u until (Q+diag(u)) is positive semidefinite. Then, using the fact that x 2 i = x i , we can obtain an equivalent convex objective function, which can then be handled by an MIQP solver. Hence, computing a suitable vector u constitutes a preprocessing phase in this exact solution method. We devise two different preprocessing methods. The first one is straightforward and consists in computing the smallest eigenvalue of Q. In the second method, vector u is obtained once a classical SDP relaxation of (P ) is solved.We carry out computational tests using the generator of (Pardalos and Rodgers, 1990) and we compare our two solution methods to several other exact solution methods. Furthermore, we report computational results for the max-cut problem.
Abstract. Let (M QP ) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (M QP ), i.e. we reformulate (M QP ) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of (M QP ). Computational experiences are carried out with instances of (M QP ) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in one hour of CPU time by a standard solver.
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