Semidefinite relaxations of chance constrained algebraic problems

In this paper, we present preliminary results on a general approach to chance constrained algebraic problems. In this type of problems, one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally complex. With the objective of developing systematic numerical procedures to solve such problems, a sequence of convex relaxations is provided, whose optimal value is shown to converge to solution of the original problem. In other words, we provide a sequence of semidefinite programs of increasing dimension and complexity which can arbitrarily approximate the solution of the probability maximization problem. Two numerical examples are presented to illustrate preliminary results on the numerical performance of the proposed approach.