Dissertation Defense by Stewart Curry

Title: Statistical Inference for Optimization Models: Sensitivity Analysis and Uncertainty Quantification

Advisor: Dr. Nicoleta Serban

Committee members (ordered alphabetically):

Dr. Roshan Joseph

Dr. Pinar Keskinocak

Dr. Ilbin Lee (University of Alberta)

Dr. Arkadi Nemirovski

Date and Time: Friday, August 23rd, 11:00 am

Location: Groseclose 402

Abstract:

In recent years, the optimization, statistics and machine learning communities have built momentum in bridging methodologies across domains by developing solutions to challenging optimization problems arising in advanced statistical modeling. While the field of optimization has contributed with general methodology and scalable algorithms to modern statistical modeling, fundamental statistics can also bring established statistical concepts to bear into optimization. In the operations research literature, sensitivity analysis is often used to study the sensitivity of the optimal decision to perturbations in the input parameters. Providing insights about how uncertain a given optimal decision might be is a concept at the core of statistical inference. Such inferences are essential in decision making because in some cases they may suggest that more data need to be acquired to provide stronger evidence for a decision; in others, they may prompt not making a decision at all because of the high uncertainty of the decision environment. Statistical inference can provide additional insights in decision making by quantifying how uncertainty in input data propagates into decision making.

In this dissertation, we propose a methodological and computational framework for statistical inference on the decision solutions derived using optimization models, particularly, high-dimensional linear programming (LP). In Chapter 2, we explore the theoretical geometric properties of critical regions, an important concept from classical sensitivity analysis and parametric linear programming, and suggest a statistical tolerance approach to sensitivity analysis which considers simultaneous variation in the objective function and constraint parameters. Using the geometric properties of critical regions, in Chapter 3, we develop an algorithm that solves LPs in batches for sampled values right-hand-side parameters (i.e. b of Ax = b in the constraints). Moreover, we suggest a data-driven version of our algorithm that uses the distribution of the b's and empirically compare our approach to other methods on various problem instances. Finally, in Chapter 4, we suggest a unified framework for statistical inference on the decision solutions and implement the framework to making statistical inferences on spatial disparities in access to dental care services.