Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

Abstract

A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables Y ij to represent each of the products x i x j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y − xx T 0 and disjunctive programming. On the other hand, their main drawback is their huge size, even for problems of moderate size. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on the box QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 sec even for larger instances with 100 variables; the SDP+RLT relaxations of the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.

Faster, but weaker, relaxations for quadratically constrained quadratic programs

“…Closely related approaches have recently been developed by Saxena et al [17] and Zheng et al [20]. In [17], the authors study the relaxation obtained by the following spectral splitting of A i :…”

“…Others have studied different types of relaxations, for example, ones based on linear programming [10,13,18] or second-order cone programming (SOCP) [9,17,20]. Generally speaking, one would expect such relaxations to provide weaker bounds in less time compared to SDP relaxations.…”

Disjunctive Inequalities: Applications And Extensions

“…Saxena, Bonami and Lee [50,51] proposed two classes of disjunctive cuts for QCQP problems, which contain constraints of the type x ⊤ Qx + b ⊤ x + c ≤ 0, i.e., nonconvex because in general Q 0. These problems are reformulated as linear programs with an extra nonconvex constraint of the form Y = xx ⊤ , where Y is an n×n matrix of auxiliary variables.…”

“…Moreover, Saxena, Bonami and Lee [50,51] used the nonconvex constraint xx ⊤ − Y 0 to obtain disjunctive cuts as follows. Disjunctions are derived from the nonconvex constraint (v ⊤ x) 2 ≥ v ⊤ Y v, where vector v is obtained from the negative eigenvalues of the matrixxx ⊤ −Ȳ , and (x,Ȳ ) is a solution of the relaxation.…”

“…Disjunctions are derived from the nonconvex constraint (v ⊤ x) 2 ≥ v ⊤ Y v, where vector v is obtained from the negative eigenvalues of the matrixxx ⊤ −Ȳ , and (x,Ȳ ) is a solution of the relaxation. In the second paper [50], this procedure is refined to generate cuts for the non-reformulated problem, thus avoiding the need of the auxiliary matrix variable Y .…”