Combining Semidefinite and Polyhedral Relaxations for Integer Programs

We present a general framework for designing semidefinite relaxations for constrained 0–1 quadratic programming and show how valid inequalities of the cut-polytope can be used to strengthen these relaxations. As examples we improve the ϑ-function and give a semidefinite relaxation for the quadratic knapsack problem. The practical value of this approach is supported by numerical experiments which make use of the recent development of efficient interior point codes for semidefinite programming.

[1]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[2]  Michael Jünger,et al.  Experiments in quadratic 0–1 programming , 1989, Math. Program..

[3]  B. Mohar,et al.  Eigenvalues and the max-cut problem , 1990 .

[4]  Caterina De Simone,et al.  The cut polytope and the Boolean quadric polytope , 1990, Discret. Math..

[5]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[6]  Egon Balas,et al.  Polyhedral methods for the maximum clique problem , 1994, Cliques, Coloring, and Satisfiability.

[7]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[8]  Charles Delorme,et al.  Laplacian eigenvalues and the maximum cut problem , 1993, Math. Program..

[9]  R. Vanderbei,et al.  An Interior-point Method for Semideenite Programming an Interior-point Method for Semideenite Programming , 1994 .

[10]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[11]  C. Helmberg.  An Interior Point Method for Semidefinite Programming and Max-Cut Bounds , 1994 .

[12]  S. Poljak,et al.  On a positive semidefinite relaxation of the cut polytope , 1995 .

[13]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[14]  Robert J. Vanderbei,et al.  An Interior-Point Method for Semidefinite Programming , 1996, SIAM J. Optim..

[15]  Franz Rendl,et al.  Quadratic Knapsack Relaxations Using Cutting Planes , 1996, IPCO.

[16]  Franz Rendl,et al.  Connections between semidefinite relaxations of the max-cut and stable set problems , 1997, Math. Program..