Interior Point Algorithms for Stochastic Semidefinite Programming
Two-stage stochastic semidefinite programming with recourse (SSDP) has been proposed and studied during the last 10 years, as a two-stage stochastic counterpart of semidefinite programming (SDP). To design efficient algorithms for solving SSDP, especially larger instances with many scenarios, it is crucial to exploit the two-stage block structure of the problem. In this dissertation, we start by the duality theory of SSDP, which illustrates that SSDP in the primal standard form and the dual standard form are not equivalent, mainly because of different block structures. Therefore algorithms must be designed separately for both forms. Homogeneous self-dual algorithm for SSDP in primal standard form was designed by Jin, Ariyawansa and Zhu . We extend their strategy to design homogeneous self-dual algorithm for the dual standard form. Infeasible path following primal-dual algorithms for SSDP in both standard forms are also developed. We further provide implementations of these algorithms. Numerical results show the advantages of these algorithms, compared to the straightforward strategy of solving deterministic equivalents of SSDP by off-the-shelf interior point solvers. Finally, we discuss another application of SSDP for solving the two-period optimal power flow problems.