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SIAM Journal on Optimization

On Nonconvex Quadratic Programming with Box Constraints

Samuel Burer, Adam N. Letchford

Published Jan 1, 2009• 75 citations
Publisher

Abstract

Nonconvex quadratic programming with box constraints is a fundamental $\mathcal{NP}$-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.

DOI
10.1137/080729529

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Nonconvex quadratic programming with box constraints is a fundamental NP\mathcal{NP}NP-hard global optimization problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterize their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral combinatorics. Finally, we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality.
Abstract Cutting planes from the Boolean Quadric Polytope can be used to reduce the optimality gap of the NP\mathcal {NP}NP NP -hard nonconvex quadratic program with box constraints (BoxQP). It is known that all cuts of the Chvátal–Gomory closure of the Boolean Quadric Polytope are A-odd cycle inequalities. We obtain a compact extended relaxation of allA-odd cycle inequalities, which allows to optimize over the Chvátal–Gomory closure without repeated calls to separation algorithms and has less inequalities than the formulation provided by Boros et al. (SIAM J Discrete Math 5(2):163–177, 1992) for sparse matrices. In a computational study, we confirm the strength of this relaxation and show that we can provide very strong bounds for the BoxQP, even with a plain linear program. The resulting bounds are significantly stronger than these from Bonami et al. (Math Program Comput 10(3):333–382, 2018), which arise from separating A-odd cycle inequalities heuristically.
Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.
We consider problem (QP) of minimizing a quadratic function subject to linear or quadratic constraints. Variables are integer and bounded. This very general problem can model many classical problems in Combinatorial Optimization.A major difference between (QP) and integer linear programs lies in the fact that, in general, its continuous relaxation is an NP-hard optimization problem. To overcome this difficulty, the Convex Quadratic Reformulation approach transforms (QP) into a problem (QP'), equivalent to (QP), but whose continuous relaxation is a convex problem. To compute a global optimal solution to (QP) it becomes possible to solve (QP') by an implicit enumeration algorithm based on continuous convex optimization.We make an overview of recent developments in this approach. We show in particular how positive semidefinite relaxations can be used to build the most interesting equivalent problems (QP'). In the case of binary variables, we give a vision of classical linearization as a special case of Convex Quadratic Reformulation.Finally, we illustrate the generality and the computational efficiency of the Quadratic Convex Reformulation approach for several combinatorial optimization problems.
ABSTRACT Quadratically constrained quadratic programming (QCQP) problems appear in a wide range of engineering fields, including computer science, communication engineering, and finance. A key difficulty in solving these problems lies in efficiently finding global solutions, especially for large-scale and nonconvex instances. To address this, Wen & Yin (2013) proposed a method that reformulates semidefinite programming (SDP) relaxations of QCQPs into a nonlinear and nonconvex low-rank problem. This reformulated problem can be efficiently solved using a curvilinear search method combined with Barzilai-Borwein (BB) steps, known as the CSBB algorithm. In this study, we compare two approaches for solving QCQPs: the conventional convex SDP relaxation and the nonconvex low-rank reformulation introduced by Wen and Yin. We propose a set of general QCQP models that are compatible with the low-rank framework and conduct a series of numerical experiments to evaluate their performance. This study evaluates performance using several classes of NP-hard problems, including Binary Integer Quadratic (BIQ) problems, Max-Cut problems, Boolean Least Squares (BLS) problems, and 0-1 Quadratic Knapsack Problems (QKP). The results demonstrate that the low-rank approach offers competitive performance and shows strong potential for solving large-scale QCQPs more efficiently than traditional convex methods.
On the Ω(n)Ω(n)Ω(n) SDP relaxation gap for Submodular Box-Constrained Quadratic Programming
Continuous submodular optimization is an important class of nonconvex problems with global guarantees, for which the submodular box-constrained quadratic programming (BCQP) forms a fundamental subclass. It arises naturally as quadratic subproblems of general twice-differentiable submodular problems and also has direct business and management applications such as pricing. This paper studies the worst-case relaxation gap of semidefinite programming (SDP) relaxations with a finite family of valid linear cuts for submodular BCQP. We show that in dimension n≥4n\geq4n≥4, for any SDP relaxation formulated with a finite family of linear cuts, there always exists a submodular BCQP instance with a strictly positive relaxation gap. By restricting to the normalized BCQP instances that remove the problem's scaling factor, we further quantify this limitation with explicit lower bounds. In general dimension n≥4n\geq4n≥4, when the SDP relaxation is formulated with only Boolean-quadric-polytope (BQP) valid cuts, including the commonly used McCormick and triangle inequalities, we establish an Ω(n)Ω(n)Ω(n) relaxation gap lower bound. For an arbitrary family of mmm valid linear cuts, we establish an Ω(n/m2)Ω(n/m^2)Ω(n/m2) lower bound. These results capture the fundamental limitations of SDP relaxations, especially SDP with BQP valid cuts, and provide guidance on what additional constraints may be needed for more effective convex approximations of submodular BCQP.