Self-concordant Tree and Decomposition Based Interior Point Methods for Stochastic Convex Optimization Problem

We consider barrier problems associated with two and multistage stochastic convex optimization problems. We show that the barrier recourse functions at any stage form a selfconcordant family with respect to the barrier parameter. We also show that the complexity value of the first stage problem increases additively with the number of stages and scenarios. We use these results to propose a prototype primal interior point decomposition algorithm for the two-stage and multistage stochastic convex optimization problems admitting self-concordant barriers.