On duality gap in binary quadratic programming

Abstract

Abstract. We present in this paper new results on the duality gap between the binary quadratic optimization problem and its Lagrangian dual or semidefinite programming relaxation. We first derive a necessary and sufficient condition for the zero duality gap and discuss its relationship with the polynomial solvability of the primal problem. We then characterize the zeroness of the duality gap by the distance, δ, between {−1, 1} n and certain affine subspace C and show that the duality gap can be reduced by an amount proportional to δ 2 . We finally establish the connection between the computation of δ and cell enumeration of hyperplane arrangement in discrete geometry and further identify two polynomially solvable cases of computing δ.

The unconstrained binary quadratic programming problem: a survey

“…Zheng, Sun, Li, and Xu (2012) present new sufficient conditions for verifying zero duality gap in nonconvex constrained quadratic programs and then show how the results specialize for UBQP. In related work, Sun, Liu, and Gao (2012) investigate the duality gap between UBQP and its semi-definite programming relaxation. Making the connection between the duality gap and the cell enumerations of hyperplane arrangement in discrete geometry, estimates of the duality gap can be derived, yielding improved lower bounds for UBQP.…”

Modern optimization theory and applications

“…对于一般的 0-1 二次优化问题, 学者们提 出了求解精确解的多种方法, 包括基于代数的方法、线性化方法、割平面算法和分支定界算法等. 文 献 [370,378] [362,379,380]). [327] 使用上述半定规划 松弛给出最大割问题 0.8756 的近似界.…”

“…在实 际应用半定规划方法中, 文献 [379] 研究了松弛问题和原问题对偶间隙的估计问题. Sun 等 [380] 提出通 过胞元枚举 (cell enumeration) 的算法可以减小半定规划产生的对偶间隙, 从而提升由半定规划得到可 行解的质量. 由于求解半定规划的代价较大, Kim 和 Kojima [382] 研究了使用二阶锥优化放松 (BQP) 问题的模型.…”

Convex reformulation for binary quadratic programming problems via average objective value maximization

“…For example, Anstreicher shows that incorporating the RLT inequality constraints into the semidefinite relaxation problem can improve the SDR bound [2]. Sun et al [16] show that by estimating the gap between the SDR problem and the original problem, the SDR bound can be further improved. Malik et al propose a spectral decomposition based method to estimate the relaxation gap for improving the lower bound [10].…”