Cutting plane methods for probabilistically-robust feasibility problems

Many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution vector x that satisfies a set of inequalities of the form F={f(x,delta) <= 0}. This set typically contains an infinite and uncountable number of inequalities, and it has been proved that the related robust feasibility problem is numerically hard to solve in general. In this paper, we discuss a family of cutting plane methods that solve efficiently a probabilistically-relaxed version of the problem. Specifically, under suitable hypotheses, we show that a cutting plane scheme based on a probabilistic oracle returns in a finite and pre-specified number of iterations a solution which is feasible for most of the members of F, except possibly for a subset having arbitrarily small probability measure.