Mixed-Integer Programming Approaches for Risk-Averse Multicriteria Optimization

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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. Despite their ubiquity and wide-ranging impact, risk-averse optimization problems that involve multiple criteria, discrete decisions and recourse actions are not well studied. The goal of this project is to bridge this gap by developing novel models and effective methods that will enhance our knowledge base, and enable the solution of more realistic and general risk-averse multicriteria optimization problems. In particular, the problems that will be studied are prevalent in homeland security and humanitarian relief applications. By developing models and effective methods for such problems, this project will improve our ability to prepare for and respond to national emergencies.This project involves three major thrusts that will advance the development of models and methods for risk-averse multicriteria optimization. First, in the realm of single-stage optimization, a novel model for risk-averse multiobjective optimization is proposed, where the relative importance of the multiple criteria is ambiguous. To this end, a new robust multivariate risk measure with desirable theoretical properties must be defined. In this project, the problem of optimizing such a risk measure will be formulated as a concave minimization problem for which effective solution methods will be developed. Second, a rigorous study of the fundamental non-convex polyhedral substructures arising in the cut generation problems for optimization under multivariate risk will be conducted. Third, two-stage optimization problems under multivariate risk will be formulated to allow recourse decisions. For these problems, decomposition algorithms that involve successive linear approximations of the second-stage problems will be devised.