Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem

Abstract

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.

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

“…The currently strongest results on these graphs are due to Billionnet and Elloumi [11] (details about their algorithm are given in Section 3). They are not able to solve instances G −1/0/1 of size n = 100 at all.…”

“…Billionnet and Elloumi [11] consider the following relaxation of (QP). Define for any vector u ∈ R n the Lagrangian…”

“…In [11] it is observed that the dual to this SDP is essentially equivalent to the basic Max-Cut relaxation (10), see Section 3.2. The solution of problem (QP u ) (or (QP u * ), respectively) can be derived by using a solver for convex quadratic 0-1 problems, i.e., a Branch-and-Bound algorithm using β(u), the continuous relaxation of QP u , as a bound.…”

“…We have chosen all the problems of sizes of our interest, which are the data sets bqpgka, due to [21] and bqp100 and bqp250, see [9]. Furthermore, in [11] the sets c and e of bqpgka are extended. We call these instances bqpbe.…”

“…The recent work of Billionnet and Elloumi [11] based on convex quadratic optimization, see Section 3.3.…”