Chance-Constrained Linear Matrix Inequality Optimization: Theory and Applications

In this dissertation, we investigate chance-constrained linear matrix inequality (LMI) optimization problems that require stochastic LMI constraints to be satisfied with high probability. While chance-constrained linear matrix inequality (CCLMI) optimization provides a natural way to model optimization problems with uncertainty, finding exact solutions to these problems is notoriously challenging due to the nonconvexity of their feasible set and requirements of high-dimensional integrations. As a result, the main goal of this dissertation is to develop computationally efficient reformulation and approximation approaches along with algorithmic methods that would enable the solution of CCLMI problems. More precisely, the research carries over: (i) the two main types of chance-constrained linear matrix inequality (CCLMI) optimization problems: CCLMI with random technology matrix (RTM), and CCLMI with random right-hand side (RHS), and (ii) the two main types of probability distributions: continuous and discrete probability distributions. For continuous probability distributions, we primarily rely on the partial sample average approximation (PSAA) method to derive a series of computationally tractable approximations for CCLMI problems, analyze their properties, and derive sufficient conditions ensuring convexity for the two most popular - normal and uniform - continuous distributions. In addition, we derive several semidefinite programming PSAA-reformulations efficiently solved by off-the-shelf solvers and design a sequential convex approximation method for the PSAA formulations containing bilinear matrix inequalities. For discrete probability distributions, we use and expand the Boolean reformulation to mixed-integer semi-definite programming (MISDP) reformulations/approximations for CCLMI problems with RTM and RHS, respectively. Furthermore, we propose different efficient Branch-and-Cut algorithms to solve mixed-integer SDP problems. We implement the reformulation, algorithmic, and theoretical findings to solve CCLMI problems in three practical applications: Robust Truss Topology design, and Optimal Control for Uncertain Linear Time-delay Systems by State Feedback Controllers with Memory, and Military Attack Problem with Mobile and Immobile targets.Release after 01/21/202