An iterative scheme for valid polynomial inequality generation in binary polynomial programming

  • Authors:

  • Bissan Ghaddar;Juan C. Vera;Miguel F. Anjos

  • Affiliations:

  • Department of Management Sciences, University of Waterloo, Waterloo, ON, Canada;Tilburg School of Economics and Management, Tilburg University, Tilburg, The Netherlands;Département de Mathématiques et génie industriel & GERAD, École Polytechnique de Montréal, Montréal, QC Canada

  • Venue:
  • IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
  • Year:
  • 2011

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Abstract

Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves the semidefinite relaxations without incurring exponential growth in their size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems. For binary polynomial programs, we prove that the proposed scheme converges to the global optimal solution for interesting cases of the initial approximation of the problem. We also present examples illustrating the computational behaviour of the scheme and compare it to other methods in the literature.