Consensus Distributionally Robust Optimization With Phi-Divergence

We study an efficient algorithm to solve the distributionally robust optimization (DRO) problem, which has recently attracted attention as a new paradigm for decision making in uncertain situations. In traditional stochastic programming, a decision is sought that minimizes the expected cost over the probability distribution of the unknown parameters. In contrast, in DRO, robust decision making can be derived from data without assuming a probability distribution; thus, it is expected to provide a powerful method for data-driven decision making. However, it is computationally difficult to solve the DRO problem and even by state-of-art solvers the problem size that can be solved to optimality is still limited. Therefore, we propose an efficient algorithm for solving DRO based on consensus optimization (CO). CO is a distributed algorithm in which a large-scale problem is decomposed into smaller subproblems. Because different local solutions are obtained by solving subproblems, a consensus constraint is imposed to ensure that these solutions are equal, thereby guaranteeing global convergence. We applied the proposed method to linear programming, quadratic programming, and second-order cone programming in numerical experiments and verified its effectiveness.