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Mathematics of Operations Research

Duality Gap Estimation of Linear Equality Constrained Binary Quadratic Programming

Xiaojin Zheng, Xiaoling Sun, Duan Li, Yong Xia

Published Nov 1, 2010• 11 citations
Publisher

Abstract

We investigate in this paper the Lagrangian duality properties of linear equality constrained binary quadratic programming. We derive an underestimation of the duality gap between the primal problem and its Lagrangian dual or SDP relaxation, using the distance from the set of binary integer points to certain affine subspace, while the computation of this distance can be achieved by the cell enumeration of hyperplane arrangement. Alternative Lagrangian dual schemes via the exact penalty and the squared norm constraint reformulations are also discussed.

DOI
10.1287/moor.1100.0472

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We investigate in this paper the Lagrangian duality properties of linear equality constrained binary quadratic programming. We derive an underestimation of the duality gap between the primal problem and its Lagrangian dual or SDP relaxation, using the distance from the set of binary integer points to certain affine subspace, while the computation of this distance can be achieved by the cell enumeration of hyperplane arrangement. Alternative Lagrangian dual schemes via the exact penalty and the squared norm constraint reformulations are also discussed.
This repository contains the source code, computational experiments, analysis, and benchmarks accompanying the paper A Direct SDP-Based Framework for Certified Bounds for Constrained Quadratic Binary Optimization Problems. The repository implements a direct semidefinite programming (SDP) approach for constrained quadratic binary optimization, designed to obtain strong bounds for exact optimization. The SDP relaxation is strengthened using quadratized linear constraints and triangle (hypermetric) inequalities. To address the numerical infeasibility of SDP solutions obtained from numerical solvers such as MOSEK, the framework includes a correction procedure that converts approximate SDP solutions into mathematically certified bounds suitable for use in Branch-and-Bound algorithms. The repository also implements an adaptation of the Goemans–Williamson randomized hyperplane rounding scheme to constrained binary optimization, together with problem-specific feasibility restoration. Computational experiments are provided for the Sparsest k-Subgraph and Maximum Independent Set problems. The work presented in this repository was funded and developed under the QBIQ project (project no. 644), funded by the Slovenian Research and Innovation Agency (ARIS) through the Applied Projects programme. The code and accompanying data are intended to facilitate reproducibility of the computational results and provide a foundation for further research on exact SDP-based methods for constrained quadratic binary optimization.
Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.
As a classical combinatorial problem, the binary quadratic programming problem has many applications in finance, statistics, production management, etc. The state-of-the-art solution for solving this problem accurately is based on branch-and-bound frameworks, with the low bound support of a semi-definite programming (SDP) relaxation. This paper generalizes the spectral bounds in the literature and proposes a sequence of improved SDP bounds for the binary quadratic programming problem. Our method relies on the closest binary points to an affine space, which can be found by reverse enumeration technique.
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.