A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming

Ariyawansa and Zhu [5] have recently proposed a new class of optimization problems termed stochastic semidefinite programs (SSDP’s). SSDP’s may be viewed as an extension of two-stage stochastic (linear) programs with recourse (SLP’s). Zhao [25] has derived a decomposition algorithm for SLP’s based on a logarithmic barrier and proved its polynomial complexity. Mehrotra and Özevin [17] have extended the work of Zhao [25] to the case of SSDP’s to derive a polynomial logarithmic barrier decomposition algorithm for SSDP’s. An alternative to the logarithmic barrier is the volumetric barrier of Vaidya [20]. It has been observed [9] that certain cutting plane algorithms [15] for SLP’s based on the volumetric barrier perform better in practice than similar algorithms based on the logarithmic barrier. There is no work based on the volumetric barrier analogous to that of Zhao [25] for SLP’s or to the work of Mehrotra and Özevin [17] for SSDP’s. The purpose of this paper is to derive a class of volumetric barrier decomposition algorithms for SSDP’s, and to prove polynomial complexity of certain members of the class. AMS Subject Classifications: 90C15, 90C51, 49M27