Parallel Computing on Semidefinite Programs

This paper demonstrates how interior-point methods can use multiple processors efficiently to solve large semidefinite programs that arise in VLSI design, control theory, and graph coloring. Previous implementations of these methods have been restricted to a single processor. By computing and solving the Schur complement matrix in parallel, multiple processors enable the faster solution of medium and large problems. The dual-scaling algorithm for semidefinite programming was adapted to a distributedmemory environment and used to solve medium and large problems than faster than could previously be solved by interior-point algorithms. Three criteria that influence the parallel scalability of the solver are identified. Numerical results show that on problems of appropriate size and structure, the implementation of an interior-point method exhibits good scalability on parallel architectures.

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

[2]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.

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

[4]  F. Potra,et al.  Interior-point methods for semidefinite programming , 1997 .

[5]  Robert A. van de Geijn,et al.  Using PLAPACK - parallel linear algebra package , 1997 .

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

[7]  Robert A. van de Geijn,et al.  PLAPACK Parallel Linear Algebra Package Design Overview , 1997, International Conference on Software Composition.

[8]  Renato D. C. Monteiro,et al.  An Efficient Algorithm for Solving the MAXCUT SDP Relaxation , 1998 .

[9]  Brian Borchers,et al.  SDPLIB 1.1, A Library of Semidefinite Programming Test Problems , 1998 .

[10]  Jon M. Kleinberg,et al.  The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover , 1998, SIAM J. Discret. Math..

[11]  S. Benson,et al.  Mixed linear and semidefinite programming for combinatorial and quadratic optimization , 1999 .

[12]  M. Todd.  A study of search directions in primal-dual interior-point methods for semidefinite programming , 1999 .

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

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

[15]  R. Saigal,et al.  An Incomplete Cholesky Factorization for Dense Symmetric Positive Definite Matrices , 2000 .

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

[17]  William Gropp,et al.  PETSc 2.0 users manual , 2000 .

[18]  Yinyu Ye,et al.  Solving Sparse Semidefinite Programs Using the Dual Scaling Algorithm with an Iterative Solver , 2000 .

[19]  Steven J. Benson,et al.  DSDP4 { A Software Package Implementing the Dual-Scaling Algorithm for Semidenite Programming 1 , 2002 .

[20]  Yin Zhang,et al.  Digital Object Identifier (DOI) 10.1007/s101070100279 , 2000 .

[21]  Kim-Chuan Toh,et al.  Solving Some Large Scale Semidefinite Programs via the Conjugate Residual Method , 2002, SIAM J. Optim..

[22]  Renato D. C. Monteiro,et al.  A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization , 2003, Math. Program..

[23]  M. Kocvara.  A Generalized Augmented Lagrangian Method for Semidefinite Programming , 2003 .