Elsevier

Operations Research Letters

Volume 39, Issue 5, September 2011, Pages 297-300
Operations Research Letters

Gap inequalities for non-convex mixed-integer quadratic programs

https://doi.org/10.1016/j.orl.2011.07.002Get rights and content

Abstract

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general non-convex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.

Highlights

► The gap inequalities form a very general class of cutting planes for the max-cut problem. ► We extend them to the case of non-convex mixed-integer quadratic programs. ► Our inequalities dominate some inequalities arising from a natural semidefinite relaxation.

Introduction

A popular and very powerful approach to solving NP-hard optimisation problems is to formulate them as integer or mixed-integer programs, and then derive strong valid linear inequalities, which can be used within cutting-plane or branch-and-cut algorithms (see, e.g., [7], [8]).

Laurent and Poljak [18] introduced an intriguing class of inequalities, called gap inequalities, for a combinatorial optimisation problem known as the max-cut problem. They showed that the gap inequalities not only dominate some inequalities arising from the well-known semidefinite programming (SDP) relaxation of the max-cut problem, but also include many other known inequalities as special cases.

In this paper, we show that the idea underlying the gap inequalities can be adapted, in a natural way, to yield gap inequalities for non-convex Mixed-Integer Quadratic Programs (MIQPs). Following Laurent and Poljak, we show that these inequalities dominate some inequalities arising from a natural SDP relaxation of non-convex MIQPs. This leads us to conjecture that the generalised gap inequalities are likely to make useful cutting planes for such problems, provided that effective heuristics for generating them can be developed.

The structure of the paper is as follows. In Section 2, we review the relevant literature. In Section 3, we derive gap inequalities for unconstrained 0–1 quadratic programs. Then, in Section 4, we derive them for general non-convex MIQPs.

Section snippets

Literature review

For surveys on the max-cut problem and related problems, we refer the reader to [11], [16]. Here, we present only what is needed for the sake of exposition.

A set F of edges in an undirected graph is called an edge cutset, or simply cut, if there exists a set S of vertices such that an edge is in F if and only if exactly one of its end-vertices is in S. It is known that a vector y{0,1}(n2) is the incidence vector of a cut in the complete graph Kn if and only if it satisfies the following

From max-cut to unconstrained 0–1 QP

Given any vector α=(α1,,αn+1)TRn+1, one can form a psd inequality for CUTn+1. Now, if the covariance mapping is applied to the psd inequality, one obtains a valid inequality for BQPn that can be written in the following form: i=1nαi(αiσ(α))xi+21i<jnαiαjXij+σ(α)2/40. These valid inequalities for BQPn were also derived by Sherali and Fraticelli [26] in a different way, using the well-known fact [21] that the matrix (1x)(1x)T=(1xTxxxT) is psd.

Now suppose that αZn+1 and σ(α) is odd. Then,

An extension to non-convex MIQP

A Mixed-Integer Quadratic Program (MIQP) is an optimisation problem of the form: min{xTQx+cTx:Axb,xiZ+(iI),xiR+(iC)}, where x is the vector of decision variables, Q is the matrix of quadratic cost terms, c is the vector of linear profit terms, Axb is a system of linear inequalities, I is the set of integer-constrained variables, and C is the set of continuous variables. We let n denote |IC|.

When the objective function is non-convex (i.e., when Q is not psd), even solving the continuous

Acknowledgments

The second and third authors were supported by the Engineering and Physical Sciences Research Council (EPSRC), under grants EP/F033613/1 and EP/D072662/1, respectively.

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