Date of Degree
PhD (Doctor of Philosophy)
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 rising 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.
Recently, stochastic programming and decision making under conditions of uncertainty have been receiving an increasing amount of attention in the literature. With the ongoing advances in the amount of computational power, it is now possible to successfully solve optimization problems in the presence of random parameters for many practical applications. In the present work two challenges associated with the introduction of randomness into optimization are discussed: how these uncertainties can be modeled, and then how the resulting problems can be solved numerically. Efforts in designing appropriate ”measures of risk” are outlined in the first chapter, with special consideration given to the phenomena of heavy-tailed distributions of losses and catastrophic risk. This leads to the introduction of a new general modeling framework that have not been considered in the literature before. Next, the mathematical programming consequences of the proposed modeling approaches are considered. This work includes design of novel solution procedures for both convex and mixed-integer programming problems of a special kind.
publicabstract, mathematical programming, mixed-integer optimization, nonlinear optimization, risk analysis, stochastic programming, uncertainty quantification
xii, 184 pages
Includes bibliographical references (pages 178-184).
Copyright 2015 Alexander Vinel