Sampled Convex Programs and Probabilistically Robust Design

Abstract

This chapter deals with the sampled scenarios approach to robust convex programming and its applications to control analysis and synthesis. It has been shown in previous work [71] that by randomly sampling a sufficient number of constraints among the (possibly) infinite constraints of a robust convex program, one obtains a standard convex optimization problem whose solution is ‘approximately feasible,’ in a probabilistic sense, for the original robust convex program. This is a generalization property in the learning theoretic sense, since the satisfaction of a certain number of ‘training’ constraints entails the satisfaction of other ‘unseen’ constraints. In this contribution we provide a new efficient bound on the generalization rate of sampled convex programs, and discuss several applications of this paradigm to robust control analysis and design problems.