Improved estimation of duality gap in binary quadratic programming using a weighted distance measure

• Published in: European journal of operational research Vol. 218; no. 2; pp. 351 - 357
• Main Authors: Xia, Yong, Sheu, Ruey-Lin, Sun, Xiaoling, Li, Duan
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16.04.2012; Elsevier; Elsevier Sequoia S.A
• Subjects: Applied sciences; Cell enumeration and hyperplane arrangement; Estimating techniques; Exact sciences and technology; Lagrange multiplier; Lagrangian duality gap; Mathematical programming; Operational research and scientific management; Operational research. Management science; Optimization algorithms; Quadratic binary programming; Quadratic programming; Semidefinite relaxation; Studies; Weighted distance measure; Cell enumeration and hyperplane arrangement; Semidefinite relaxation; Weighted distance measure; Lagrangian duality gap; Quadratic binary programming; Enumeration; Subgradient optimization; Necessary and sufficient condition; Semi definite programming; Duality; Quadratic programming; Hyperplane; Lagrange multiplier; Zero one programming; Vector space
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2011.10.034

► An improved estimate for the duality gap of binary quadratic programming is obtained. ► We use the weighted distance measure and the cell enumeration in discrete geometry. ► The optimal choice of the weighted matrix can be found via a SDP. ► We study conditions under which the weighted measure gives a strictly tighter bound. We present in this paper an improved estimation of duality gap between binary quadratic program and its Lagrangian dual. More specifically, we obtain this improved estimation using a weighted distance measure between the binary set and certain affine subspace. We show that the optimal weights can be computed by solving a semidefinite programming problem. We further establish a necessary and sufficient condition under which the weighted distance measure gives a strictly tighter estimation of the duality gap than the existing estimations.