Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization

We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large-scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interior-point algorithms for approximating maximum cut semidefinite programs with dimension up to 3,000.

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

[2]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[3]  Mauricio G. C. Resende,et al.  An implementation of Karmarkar's algorithm for linear programming , 1989, Math. Program..

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

[5]  Franz Rendl,et al.  A computational study of graph partitioning , 1994, Math. Program..

[6]  Franz Rendl,et al.  A recipe for semidefinite relaxation for (0,1)-quadratic programming , 1995, J. Glob. Optim..

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

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

[9]  I. Duff.  Sparse numerical linear algebra: Direct methods and preconditioning , 1996 .

[10]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[11]  Satissed Now Consider.  Improved Approximation Algorithms for Maximum Cut and Satissability Problems Using Semideenite Programming , 1997 .

[12]  I. Duff,et al.  The state of the art in numerical analysis , 1997 .

[13]  R. Vanderbei.  Primal-dual Aane-scaling Algorithms Fail for Semideenite Programming , 1997 .

[14]  Masakazu Kojima,et al.  Semidefinite Programming Relaxation for Nonconvex Quadratic Programs , 1997, J. Glob. Optim..

[15]  Masakazu Kojima,et al.  Exploiting sparsity in primal-dual interior-point methods for semidefinite programming , 1997, Math. Program..

[16]  Shinji Hara,et al.  Interior-Point Methods for the Monotone Semidefinite Linear Complementarity Problem in Symmetric Matrices , 1997, SIAM J. Optim..

[17]  Michael J. Todd,et al.  Primal-Dual Interior-Point Methods for Self-Scaled Cones , 1998, SIAM J. Optim..

[18]  Qing Zhao.  Semidefinite programming for assignment and partitioning problems , 1998 .

[19]  Franz Rendl,et al.  Semidefinite Programming Relaxations for the Quadratic Assignment Problem , 1998, J. Comb. Optim..

[20]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[21]  Masakazu Muramatsu.  Affine scaling algorithm fails for semidefinite programming , 1998, Math. Program..

[22]  Masakazu Muramatsu,et al.  Primal-Dual Affine-Scaling Algorithms Fail for Semidefinite Programming , 1999, Math. Oper. Res..

[23]  Yinyu Ye,et al.  Approximating quadratic programming with bound and quadratic constraints , 1999, Math. Program..

[24]  Masakazu Kojima,et al.  Numerical Evaluation of SDPA (Semidefinite Programming Algorithm) , 2000 .

[25]  Franz Rendl,et al.  Solving Graph Bisection Problems with Semidefinite Programming , 2000, INFORMS J. Comput..

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

[27]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .