Data-driven distributionally robust chance-constrained linear matrix inequalities

Abstract

In this thesis, we present approximation and reformulation techniques for prob­lem with distributionally robust chance-constrained linear matrix inequality (DRCCLMI), aiming at overcoming the computational challenges posed by multidimensional integration and nonconvexity of feasible sets. DRCCLMI seek a robust solution which guarantees that the chance constraints are fulfilled for a wide range of possible distribution within an ambiguity set. Specifically, we consider a data-driven ambiguity set which includes all the potential distri­butions sharing the same moment information. We first propose an inner ap­proximation for DRCCLMI in a general form to deal with common constraint structures encountered in real-world scenarios. The key method we use for ap­proximation is the Conditional Value-at-Risk (CVaR) approach, which enables us to approximate DRCCLMI in a way that ensures a certain level of solution quality while maintaining the robustness of the original constraint. Second, we derive an inner approximation and exact reformulations for a special case of DRCCLMI with and without support information, respectively. Specifi­cally, this special case refers to the situation where a block matrix structure is inherent in the linear matrix inequality. Notably, our approximation and re­formulation techniques facilitate the transformation of the original DRCCLMI into a more tractable semidefinite programming (SDP) problem, simplifying the computational process and improving the accuracy and efficiency of the so­lution. The practicality and effectiveness of these techniques are demonstrated through numerical studies on two real-world applications: truss topology design problem and calibration problem.