The Bundle Method for Hard Combinatorial Optimization Problems

Solving the well known relaxations for large scale combinatorial optimization problems directly is out of reach. We use Lagrangian relaxations and solve it with the bundle method. The cutting plane model at each iteration which approximates the original problem can be kept moderately small and we can solve it very quickly. We report successful numerical results for approximating maximum cut.

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

[2]  László Lovász,et al.  Semidefinite Programs and Combinatorial Optimization , 2003 .

[3]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

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

[5]  Yurii Nesterov,et al.  New variants of bundle methods , 1995, Math. Program..

[6]  Francisco Barahona,et al.  The volume algorithm: producing primal solutions with a subgradient method , 2000, Math. Program..

[7]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

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

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

[10]  B. Mohar,et al.  Eigenvalues in Combinatorial Optimization , 1993 .

[11]  Johan Håstad,et al.  Some optimal inapproximability results , 1997, STOC '97.

[12]  Franz Rendl,et al.  Nonpolyhedral Relaxations of Graph-Bisection Problems , 1995, SIAM J. Optim..

[13]  Michel X. Goemans,et al.  Semideenite Programming in Combinatorial Optimization , 1999 .

[14]  F. Barahona.  The max-cut problem on graphs not contractible to K5 , 1983 .

[15]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[16]  Richard M. Karp,et al.  On the Computational Complexity of Combinatorial Problems , 1975, Networks.

[17]  F. Rendl.  Semidefinite programming and combinatorial optimization , 1999 .

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

[19]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[20]  F. Hadlock,et al.  Finding a Maximum Cut of a Planar Graph in Polynomial Time , 1975, SIAM J. Comput..

[21]  C. Lemaréchal,et al.  Nonsmooth optimization : proceedings of a IIASA workshop, March 28-April 8, 1977 , 1978 .

[22]  Franz Rendl,et al.  Incorporating Inequality Constraints in the Spectral Bundle Method , 1998, IPCO.

[23]  Krzysztof C. Kiwiel,et al.  Proximity control in bundle methods for convex nondifferentiable minimization , 1990, Math. Program..

[24]  Franz Rendl,et al.  A Spectral Bundle Method for Semidefinite Programming , 1999, SIAM J. Optim..

[25]  George L. Nemhauser,et al.  A polynomial algorithm for the max-cut problem on graphs without long odd cycles , 1984, Math. Program..