Gap inequalities for non-convex mixed-integer quadratic programs
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 -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 of edges in an undirected graph is called an edge cutset, or simply cut, if there exists a set of vertices such that an edge is in if and only if exactly one of its end-vertices is in . It is known that a vector is the incidence vector of a cut in the complete graph if and only if it satisfies the following
From max-cut to unconstrained 0–1 QP
Given any vector , one can form a psd inequality for . Now, if the covariance mapping is applied to the psd inequality, one obtains a valid inequality for that can be written in the following form: These valid inequalities for were also derived by Sherali and Fraticelli [26] in a different way, using the well-known fact [21] that the matrix is psd.
Now suppose that and is odd. Then,
An extension to non-convex MIQP
A Mixed-Integer Quadratic Program (MIQP) is an optimisation problem of the form: where is the vector of decision variables, is the matrix of quadratic cost terms, is the vector of linear profit terms, is a system of linear inequalities, is the set of integer-constrained variables, and is the set of continuous variables. We let denote .
When the objective function is non-convex (i.e., when 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.
References (26)
- et al.
On a positive semidefinite relaxation of the cut polytope
Linear Algebra Appl.
(1995) Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming
J. Global Optim.
(2009)- et al.
Stronger linear programming relaxations of max-cut
Math. Program.
(2003) - et al.
On the cut polytope
Math. Program.
(1986) - et al.
Experiments in quadratic 0–1 programming
Math. Program.
(1989) Dynamic Programming
(1957)- et al.
Cut-polytopes, Boolean quadric polytopes and nonnegative quadratic pseudo-Boolean functions
Math. Oper. Res.
(1993) - et al.
Polyhedral approaches to mixed-integer linear programming
Fifty-plus years of combinatorial integer programming
The cut polytope and the Boolean quadric polytope
Discrete Math.
(1989)
On the Hamming geometry of unitary cubes
Sov. Phys. Dokl.
Geometry of Cuts and Metrics
Exploring the relationship between max-cut and stable set relaxations
Math. Program.
Cited by (11)
Valid inequalities for quadratic optimisation with domain constraints
2021, Discrete OptimizationA note on representations of linear inequalities in non-convex mixed-integer quadratic programs
2017, Operations Research LettersNon-convex mixed-integer nonlinear programming: A survey
2012, Surveys in Operations Research and Management ScienceComplexity results for the gap inequalities for the max-cut problem
2012, Operations Research LettersThe Boolean Quadric Polytope
2022, The Quadratic Unconstrained Binary Optimization Problem: Theory, Algorithms, and ApplicationsImproving the linear relaxation of maximum k-cut with semidefinite-based constraints
2019, EURO Journal on Computational Optimization