CONTRIBUTIONS TO THE THEORY OF SEMIDEFINITE PROGRAMMING UNDER UNCERTAINTY

Thesis (Ph.D.), Department of Mathematics, Washington State UniversityStochastic semidefinite programs (SSDPs) were introduced recently as a natural extension of two-stage stochastic linear programs and of semidefinite programs. Theoretical results for stochastic linear programs (SLPs) and those for semidefinite programs (SDPs) have been obtained over the last 60 years by disjoint groups of researchers. Some concepts and even theories readily migrate from SLP and SDP to SSDP. However, the combination of SLP and SDP creates fundamental differences between SSDP and SLP or SDP. The notion of an equivalent convex program for SLP and its theory were developed in late 1960s. In this dissertation, we develop the equivalent convex program of a two-stage SSDP. Analysis shows that the objective function of the equivalent problem is convex and continuous. We also show that the solution set of the dual problem of the equivalent convex program is spectrahedral. Under certain circumstances, the solution set is also polyhedral. For the most possible generalized SSDP, where the coefficients of the decision variables are also random, there exists some weak continuity-type condition (W-condition) guaranteeing the problem is solvable almost surely for a selected set of first stage variable. The existence of the W-condition also leads to a unified feasible solution which could be different based on two different interpretations of the precise meaning of stochastic constraints. Finally, we show that the objective function of an SSDP with recourse is lower semicontinuous.Department of Mathematics, Washington State Universit