Huifu Xu (University of Southampton): Utility Preference Robust Optimization: Piecewise Linear Approximation and Statistical Robustness
Preference robust optimization (PRO) has recently received increasing attention in the research communities of robust optimization and decision analysis. Differing from traditional robust optimization models which deal with ambiguity of exogenous uncertainty of a decision making problem, PRO model focuses on endogenous uncertainty arising from decision maker’s ambiguity of utility preference and/or risk attitude. In this talk, we discuss a PRO model where information on decision maker’s utility preference is incomplete but can be elicited through partial information such as questionnaires and pairwise comparison, the optimal decision is based on the least favourable utility function elicited. Differing from the existing research in this area, we propose a piecewise linear approximation scheme for the elicited utility functions and then develop efficient computational schemes for solving the approximated problem. When the utility functions are concave, we can reformulate the approximated maximin problem as a single linear programming problem. The piecewise linear approximation scheme also enables us to derive an error bound for the optimal value which is a step forward from qualitative convergence results. A key assumption in the PRO model is that the true probability distribution is either known or can be recovered by empirical data which do not contain any noise. It is unclear however a statistical estimator such as the optimal value of a PRO model based on empirical data is reliable when the empirical data contain some noise. We move on to investigate the issue which is known as statistical robustness in the literature of robust statistics. We derive moderate sufficient conditions under which the robust optimal value changes continuously against small variation of the probability distribution of the underlying random variables and identify appropriate metrics under which the statistical estimator of the optimal value is uniformly asymptotically consistent which is also known as uniform Glivenko-Cantelli property. Finally, we demonstrate statistical robustness of the estimators of the optimal value and the optimal solutions for the PRO model, the results cover a wide range of utility optimization problems when the decision variable is fixed, and stochastic optimization problems when the ambiguity of the utility disappears.