Volumetric Barrier Cutting Plane Algorithms for Stochastic Linear Semi-Infinite Optimization

Abstract

In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier cutting plane interior point methods for solving this class of optimization problems, and present a complexity analysis of the proposed algorithms. We establish our convergence analysis by showing that the volumetric barrier associated with the recourse function of stochastic linear semi-infinite programs is a strongly self-concordant barrier and forms a self-concordant family on the firststage solutions. The dominant terms in the complexity expressions obtained in this paper are given in terms of the problem dimension and the number of realizations. The novelty of our algorithms lies in their ability to kill the effect of the radii of the largest Euclidean balls contained in the feasibility sets on the dominant complexity terms. INDEX TERMS Semi-infinite programming, stochastic linear programming, interior-point methods, volumetric barrier, cutting plane.