An Iterative Scheme for Valid Polynomial Inequality Generation in Binary Polynomial Programming
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.
KeywordsBinary Polynomial Programming Binary Quadratic Programming Valid Inequality Generation Semidefinite Programming
Unable to display preview. Download preview PDF.
- 9.Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem. Annals of Operations Research (to appear)Google Scholar
- 10.Ghaddar, B., Vera, J., Anjos, M.F.: Second-order cone relaxations for binary quadratic polynomial programs. SIAM Journal on Optimization (to appear)Google Scholar
- 11.Hungerländer, P., Rendl, F.: Semidefinite relaxations of ordering problems. Technical report, Alpen-Adria-Universität Klagenfurt (August. 2010)Google Scholar
- 12.Kim, S., Kojima, M., Mevissen, M., Yamashita, M.: Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion. To appear in Mathematical Programming (2009)Google Scholar
- 24.Nesterov, Y.: Structure of non-negative polynomials and optimization problems. Technical report, Technical Report 9749, CORE (1997)Google Scholar
- 27.Parrilo, P.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, Department of Control and Dynamical Systems, California Institute of Technology, Pasadena, California (2000)Google Scholar
- 28.Parrilo, P.: An explicit construction of distinguished representations of polynomials nonnegative over finite sets. Technical report, IFA Technical Report AUT02-02, Zurich - Switzerland (2002)Google Scholar
- 31.Peña, J.F., Vera, J.C., Zuluaga, L.F.: Exploiting equalities in polynomial programming. Operations Research Letters 36(2) (2008)Google Scholar
- 40.Sturm, J.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11–12 (1999)Google Scholar
- 44.Zuluaga, L.: A conic programming approach to polynomial optimization problems: Theory and applications. PhD thesis, The Tepper School of Business, Carnegie Mellon University, Pittsburgh (2004)Google Scholar
- 45.Zuluaga, L., Vera, J.C., Peña, J.: LMI approximations for cones of positive semidefinite forms. SIAM Journal on Optimization 16(4) (2006)Google Scholar