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

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  • Xia, Yong
  • Sheu, Ruey-Lin
  • Sun, Xiaoling
  • Li, Duan

Abstract

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.

Suggested Citation

  • Xia, Yong & Sheu, Ruey-Lin & Sun, Xiaoling & Li, Duan, 2012. "Improved estimation of duality gap in binary quadratic programming using a weighted distance measure," European Journal of Operational Research, Elsevier, vol. 218(2), pages 351-357.
    • Handle: RePEc:eee:ejores:v:218:y:2012:i:2:p:351-357
    DOI: 10.1016/j.ejor.2011.10.034
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    References listed on IDEAS

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    1. Ferrez, J.-A. & Fukuda, K. & Liebling, Th.M., 2005. "Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm," European Journal of Operational Research, Elsevier, vol. 166(1), pages 35-50, October.
    2. Helmberg, C., 2002. "Semidefinite programming," European Journal of Operational Research, Elsevier, vol. 137(3), pages 461-482, March.
    3. NESTEROV, Yu., 1998. "Semidefinite relaxation and nonconvex quadratic optimization," LIDAM Reprints CORE 1362, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, December.
    5. Alidaee, Bahram & Glover, Fred & Kochenberger, Gary & Wang, Haibo, 2007. "Solving the maximum edge weight clique problem via unconstrained quadratic programming," European Journal of Operational Research, Elsevier, vol. 181(2), pages 592-597, September.
    6. Xiaojin Zheng & Xiaoling Sun & Duan Li & Yong Xia, 2010. "Duality Gap Estimation of Linear Equality Constrained Binary Quadratic Programming," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 864-880, November.
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