A Barrier Lagrangian Dual Method for Multi-stage Stochastic Convex Semidefinite Optimization

In this paper, we present a polynomial-time barrier algorithm for solving multi-stage stochastic convex semidefinite optimization based on the Lagrangian dual method which relaxes the nonanticipativity constraints. We show that the barrier Lagrangian dual functions for our setting form self-concordant families with respect to barrier parameters. We also use the barrier function method to improve the smoothness of the dual objective function so that the Newton search directions can be exploited for usage. We finally implement the proposed algorithm on different test problems to show its effectiveness and efficiency.