PhD Oral Defence – Tractable Data-Driven Distributionally Robust Optimization with (in) finitely Constrained Ambiguity Sets
Abstract
Distributionally robust optimization (DRO) has become an attractive approach for addressing optimization problems involving decision making under uncertainty. In DRO, the underlying distribution of uncertainty is ambiguous but resides in a family called the ambiguity set, the introduction of which leads to great a great modeling power in describing distributional information about uncertainty.
This thesis proposes a new class of infinitely constrained ambiguity sets, whose key feature is the potential inclusion of an infinite number of expectation constraints in its description. We elucidate the generality of this class of ambiguity sets for specifying distributional ambiguity and show its capability in characterizing several kinds of dominance relations. Although the corresponding DRO problem with this class of ambiguity sets may not necessarily lead to tractable reformulations, we approach the problem with an algorithm that solves a sequence of tractable relaxed problems. The proposed algorithm converges with the theoretical guarantee and converges reasonably well in practice.
Optimization problems across a broad range of industrial fields span along the time horizon with sequentially revealed uncertainties and require recourse decisions. The thesis also investigates DRO problems with recourse concerning with infinitely constrained ambiguity sets. Using the linear decision rule (LDR) technique, the relaxed problem admits a computationally tractable reformulation and can be iteratively refined.
Finally, the thesis presents a tractable framework for DRO. By incorporating both cone-based ambiguity sets and statistical-distance-based ambiguity sets (e.g., phi-divergence ambiguity set and Wasserstein ambiguity set) in a unified fashion, the framework is suitable for data-driven DRO. To address a DRO problem with recourse, we introduce the tractable adaptive recourse scheme (TARS), which is based on the classical LDR and can also be applied in situations where recourse decisions are discrete. A direct implementation of the framework can handle relaxed problems arising from our proposed algorithm for DRO with infinitely constrained ambiguity sets.