Linear Matrix Inequalities with Stochastically Dependent Perturbations and Applications to Chance-Constrained Semidefinite Optimization

The wide applicability of chance–constrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so–called safe tractable approximations of chance–constrained programs, where a chance constraint is replaced by a deterministic and efficiently computable inner approximation. Currently, such approach applies mainly to chance–constrained linear inequalities, in which the data perturbations are either independent or define a known covariance matrix. However, its applicability to chance– constrained conic inequalities with dependent perturbations—which arises in finance, control and signal processing applications—remains largely unexplored. In this paper, we develop safe tractable approximations of chance–constrained affinely perturbed linear matrix inequalities, in which the perturbations are not necessarily independent, and the only information available about the dependence structure is a list of independence relations. To achieve this, we establish new large deviation bounds for sums of dependent matrix–valued random variables, which are of independent interest. A nice feature of our approximations is that they can be expressed as systems of linear matrix inequalities, thus allowing them to be solved easily and efficiently by off–the–shelf solvers. We also provide a numerical illustration of our constructions through a problem in control theory.