| 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 are either independent or define a known covariance matrix. However, its applicability to the case of chance--constrained conic inequalities with dependent perturbations---which arises in supply chain management, finance, control and signal processing applications---remains largely unexplored.;In this thesis, we consider the problem of processing 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. Using large deviation bounds for matrix--valued random variables, we develop safe tractable approximations of those chance constraints. Extensions to the Matrix CVaR (Conditional Value--at--Risk) risk measure and general polynomials perturbations are also provided separately. Further more, we show that the chance--constrained linear matrix inequalities optimization problem can be converted to a robust optimization problem by constructing the uncertainty set of the corresponding robust counterpart. 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 optimization solvers. We also provide a numerical illustration of our constructions through a problem in control theory and a portfolio VaR (Value-at-Risk) optimization problem.;Key words: chance--constrained programming, dependence perturbation, safe tractable approximation, Value-at-Risk, portfolio optimization. |
