Properties and Paradigms of Robust Optimization

Robust optimization is a methodology for dealing with uncertainty in optimization problems. In this thesis, we provide a deeper understanding of the properties and paradigms of robust optimization. We examine the continuity properties of robust optimization problems with respect to their uncertainty sets, and the paradigms of implementation error under a robust optimization framework, both of which arose from questions asked during the application of robust optimization to healthcare problems. We begin with an overview of the robust optimization methodology, starting with a general standard formulation of a robust optimization problem, following with a discussion of a variety of specific well-known robust optimization formulations, and ending with a presentation of some generalizations and extensions of robust optimization. We then present a computational study verifying the effectiveness of an adaptive and robust radiation therapy (ARRT) methodology for treating lung cancer for a class of physically motivated breathing patterns, adding to the results of the original paper which proposed the methodology. Inspired by the convergence properties of the ARRT method, we establish a more general result regarding the continuity of robust optimization with respect to the uncertainty set. In particular, we prove quantitative stability of the optimal value and approximate optimal solution set and the qualitative stability of the optimal solution set, by way of a transformation that links robust linear optimization and linear semi-infinite optimization. Finally, we study the application of robust optimization to network flow problems under implementation error, with applications to manufacturing process flexibility problems. In classical process flexibility literature, it is implicitly assumed that the underlying network flow problem is always implemented perfectly, with most focus being placed on network design in order to ensure responsiveness to demand variation. In contrast, we study the underlying network flow problem, and how feasibility-preserving implementation error on the network flow solutions can be detrimental to performance of certain network designs. We formulate a robust optimization problem with an endogenous uncertainty set to model the system, but ultimately use an auxiliary linear programming problem to generate solutions which, though not provably optimal, are nonetheless generally more robust than standard solutions.Ph.D