On duality gap in binary quadratic programming

Authors:
X. L. Sun;C. L. Liu;D. Li;J. J. Gao
Affiliations:
Department of Management Science, School of Management, Fudan University, Shanghai, People's Republic of China 200433;Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, People's Republic of China 200433;Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong;Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong
Venue:
Journal of Global Optimization
Year:
2012

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Abstract

We investigate in this paper the duality gap between the binary quadratic optimization problem and its semidefinite programming relaxation. We show that the duality gap can be underestimated by $${\xi_{r+1}\delta^2}$$ , where 驴 is the distance between {驴1, 1} n and certain affine subspace, and 驴 r+1 is the smallest positive eigenvalue of a perturbed matrix. We also establish the connection between the computation of 驴 and the cell enumeration of hyperplane arrangement in discrete geometry.