Degeneracy in maximal clique decomposition for Semidefinite Programs

Exploiting sparsity in Semidefinite Programs (SDP) is critical to solving large-scale problems. The chordal completion based maximal clique decomposition is the preferred approach for exploiting sparsity in SDPs. In this paper, we show that the maximal clique-based SDP decomposition is primal degenerate when the SDP has a low rank solution. We also derive conditions under which the multipliers in the maximal clique-based SDP formulation is not unique. Numerical experiments demonstrate that the SDP decomposition results in the schur-complement matrix of the Interior Point Method (IPM) having higher condition number than for the original SDP formulation.

[1]  Katsuki Fujisawa,et al.  Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results , 2003, Math. Program..

[2]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

[3]  Kim-Chuan Toh,et al.  On the Nesterov-Todd Direction in Semidefinite Programming , 1998, SIAM J. Optim..

[4]  Charles R. Johnson,et al.  Positive definite completions of partial Hermitian matrices , 1984 .

[5]  E. D. Klerk,et al.  Aspects of semidefinite programming : interior point algorithms and selected applications , 2002 .

[6]  Masakazu Kojima,et al.  Exploiting Sparsity in SDP Relaxation of Polynomial Optimization Problems , 2012 .

[7]  Makoto Yamashita,et al.  A high-performance software package for semidefinite programs: SDPA 7 , 2010 .

[8]  Anton van den Hengel,et al.  Semidefinite Programming , 2014, Computer Vision, A Reference Guide.

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

[10]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[11]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[12]  Masakazu Kojima,et al.  Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0) , 2003, Optim. Methods Softw..

[13]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[14]  Makoto Yamashita,et al.  Fast implementation for semidefinite programs with positive matrix completion , 2015, Optim. Methods Softw..

[15]  Kim-Chuan Toh,et al.  Solving Large Scale Semidefinite Programs via an Iterative Solver on the Augmented Systems , 2003, SIAM J. Optim..

[16]  Kazuo Murota,et al.  Exploiting Sparsity in Semidefinite Programming via Matrix Completion I: General Framework , 2000, SIAM J. Optim..