Mixed-Integer Programming Approaches for Risk-Averse Multicriteria Optimization

Risk-averse optimization aims to address decision-making problems in the presence of uncertainty that involves events of low probability but severe consequences. In addition, decision makers are often required to consider multiple and conflicting performance criteria when faced with such problems. This requirement can be met by incorporating decision-based stochastic multivariate risk preferences into either (1) the constraints or (2) the objectives of the optimization models. The former class of problems, in which the random outcome vector is constrained to outperform a random benchmark vector, has received some recent attention in the literature. However, there is a paucity of research on the latter class, i.e., risk-averse multiobjective optimization. Furthermore, both classes of problems are non-convex and large-scale, because large sample sizes are necessary to capture rare events. Despite their ubiquity and wide-ranging impact, risk-averse optimization problems that involve multiple criteria, discrete decisions and recourse actions are not well studied. In this project, we propose to bridge this gap by developing novel models and effective methods that will greatly enhance our knowledge base, and enable the solution of more realistic and general risk-averse multicriteria optimization problems. Our proposed research lies at the confluence of stochastic programming, multiobjective optimization, and mixed-integer programming. These three fields have matured separately over the past few decades, but their intersection, especially under risk aversion, is much less understood. This proposal was inspired by our recent work in solving multivariate conditional value-at-risk and second-order stochastic dominance constrained optimization problems. We use an innovative representation of value-at-risk and exploit its combinatorial properties to construct effective solution methods that can handle sample sizes that are orders of magnitude larger than the best available methods today. Our preliminary results suggest great potential for future research, which we have outlined herein.