A conic IPM decomposition approach for large scale

We describe a conic interior point decomposition approach for solving a large scale semidefinite program (SDP) whose primal feasible set is bounded. The idea is to solve such an SDP using existing primal-dual interior point methods, in an iterative fashion between amaster problem and a subproblem. In our case, the master problem is a mixed conic problem over linear and smaller sized semidefinite cones. The subproblem is a smaller structured semidefinite program that either returns a column or a small sized matrix depending on the multiplicity of the minimum eigenvalue of the dual slack matrix associated with the semidefinite cone. We motivate and develop our conic decomposition methodology on semidefinite programs and also discuss various issues involved in an efficient implementation. Computational results on several well known classes of semidefinite programs are presented. Kartik K. Sivaramakrishnan: Department of Mathematics, P.O. Box 8205, North Carolina State University, Raleigh, NC 27695-8205, USA. e-mail: [email protected]. Research supported by SHARCNET and a startup grant from North Carolina State University. Gema Plaza: Department of Statistics & Operations Research, University of Alicante, Apartado de Correos 99, 03080 Alicante, Spain. e-mail: [email protected]. Research supported by Grant CTBPRB/2003/165 from FPI Program of GVA, Spain. Tamás Terlaky: Advanced Optimization Laboratory, Department of Computing & Software, McMaster University, Hamilton, Ontario L8S 4K1, Canada. e-mail: [email protected]. Research supported by NSERC, Canada Research Chair Program, and MITACS. Mathematics Subject Classification (1991): 20E28, 20G40, 20C20 Correspondence to: Kartik K. Sivaramakrishnan 2 Kartik K. Sivaramakrishnan et al.