Extreme Points of Sets of Randomized Strategies in Constrained Optimization and Control Problems

Abstract

This paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems. We first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment-like constraints. Conditions are given under which those extreme points are randomizations of at most n+1 deterministic strategies. This result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic, allocation-like, optimization problems and their Young relaxations. Similar results are obtained for constrained Markov control processes in Borel spaces.