A Probabilistic Analytic Center Cutting Plane Method for Feasibility of Uncertain LMIs

Many robust control problems can be formulated in abstract form as convex feasibility programs, where one seeks a solution x that satisfies a set of inequalities of the form F = {f(x, delta) <= 0, delta is an element of D}. 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 an Analytic Center Cutting Plane scheme based on a probabilistic oracle returns in a finite and prespecified number of iterations a solution x which is feasible for most of the members of F, except possibly for a subset having arbitrarily small probability measure.