Mathematical programming techniques for solving stochastic optimization problems with certainty equivalent measures of risk

Abstract

The problem of risk-averse decision making under uncertainties is studied from both modeling and computational perspectives. First, we consider a framework for constructing coherent and convex measures of risk which is inspired by infimal convolution operator, and prove that the proposed approach constitutes a new general representation of these classes. We then discuss how this scheme may be effectively employed to obtain a class of certainty equivalent measures of risk that can directly incorporate decision maker’s preferences as expressed by utility functions. This approach is consequently utilized to introduce a new family of measures, the log-exponential convex measures of risk. Conducted numerical experiments show that this family can be a useful tool when modeling risk-averse decision preferences under heavy-tailed distributions of uncertainties. Next, numerical methods for solving the arising optimization problems are developed. A special attention is devoted to the class p-order cone programming problems and mixed-integer models. Solution approaches proposed include approximation schemes for p-order cone and more general nonlinear programming problems, lifted conic and nonlinear valid inequalities, mixed-integer rounding conic cuts and new linear disjunctive cuts.