Title: Chance Constrained Programs and Distributionally Favorable Optimization

Virtual Link:  https://gatech.zoom.us/my/meet.weijun?pwd=NTdEVkd4cGVlck9pTEg2M0N2ZFJxZz09

Time: 2:30 pm – 4:30 pm, July 19, 2024 (Friday)

Thesis Committee Members:

Dr. Weijun Xie (Advisor), Industrial and Systems Engineering, Georgia Institute of Technology

Dr. Xin Chen, Industrial and Systems Engineering, Georgia Institute of Technology

Dr. Simge Küçükyavuz, Industrial Engineering and Management Sciences, Northwestern University

Dr. George Lan, Industrial and Systems Engineering, Georgia Institute of Technology

Dr. Alexander Shapiro, Industrial and Systems Engineering, Georgia Institute of Technology

Abstract:

This dissertation addresses the significant theoretical and computational challenges in solving chance constrained programs (CCPs), which seek optimal decisions that meet uncertain constraints within a specified risk level. Two main approaches to these challenges are: (i) developing convex inner approximations for CCPs, and (ii) solving CCPs to optimality. This dissertation improves these approaches by developing new convex approximations and creating a framework for efficient optimality cuts. Additionally, it introduces a novel distributionally favorable optimization (DFO) framework to mitigate the impact of outliers in decision-making under uncertainty.

The first part of this dissertation (Chapters 2 and 3) focuses on advancing the existing knowledge of chance constrained programs. We first study and generalize the ALSO-X, originally proposed by Ahmed, Luedtke, Song, and Xie in 2017, for solving a CCP. We show that when uncertain constraints are convex in the decision variables, ALSO-X always outperforms the CVaR approximation. We further show (i) sufficient conditions under which ALSO-X can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance ALSO-X with a convergent alternating minimization method (ALSO-X+); (iii) an extension of ALSO-X and ALSO-X+ to solve distributionally robust chance constrained programs (DRCCPs) under \infty-Wasserstein ambiguity set. While existing methods have predominantly focused on either inner or outer approximations, we bridge the gap by studying a scheme that effectively combines these approximations via variable fixing. Our empirical results showcase the advantages of our approach, both in terms of computational efficiency and solution quality.

In the second part (Chapters 4 and 5) of this thesis, we propose a novel distributionally favorable optimization (DFO) framework to mitigate the effect of outliers for decision-making under uncertainty. In cases where outliers lead to extremely large or even infinite recourse function values, the commonly used distributionally robust optimization (DRO) framework tends to overly emphasize these outliers, resulting in undesirable or even infeasible decisions. In contrast, our proposed DFO framework considers the best-case expected recourse function under the most favorable distribution from the distributional family. In this way, we show that DFO could find a proper measure of the stochastic recourse function and reduce the effect of outliers. We also show that DFO recovers many robust statistics, suggesting that the DFO framework can provide appropriate decisions in the presence of outliers. In contrast to the traditional DRO paradigm, DFO presents a unique challenge-- the application of the inner infimum operator often fails to retain the convexity. In light of this challenge, we study the tractability and complexity of DFO. We establish sufficient and necessary conditions for determining when DFO problems are tractable or intractable. Despite the typical nonconvex nature of DFO problems, our findings show that they are mixed-integer convex programming representable (MICP-R), thereby enabling solutions via standard optimization solvers.