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 an approach applies mainly to chance-constrained linear inequalities, in which the data perturbations either are 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. [PUBLICATION ABSTRACT]