This proposed research provides funding for the study of optimization problems where the uncertainty intrinsic to the constraints in the problem is modeled using a concept known as stochastic dominance. Two optimization problems which will receive an early focus in the research are the uncertain linear and uncertain semidefinite programs under a second-order linear stochastic dominance concept, which constitutes a particular way to model multi-dimensional stochastic orders. Efficient algorithms for such problems will be constructed. In addition, a duality theory that allows explicit construction of dual functions associated with the solution of such problems will be developed. More general stochastic orders and the solvability of corresponding optimization problems will be analyzed. The research will also address the situation where the support of the random entities in the stochastic dominance constraints is not finite or is very large, so that sampling based approaches are required. Finally, a study of stochastic entities with random parameters and their applications may be conducted within the context of stochastic dominance.
If successful, the proposed research will address the fundamental problem of optimizing a system where some components are not known with certainty, which has applications in many areas, including operations research, statistics and finance. The work will help to develop a better understanding of the benefits and drawbacks of using the concept of stochastic dominance --- which has proven to be of capital importance in many areas, ranging from economics to epidemiology --- in an optimization problem. One goal of this research is to develop algorithms for such problems, the availability of which will result in better modeling of parameter uncertainty in stochastic models. The proposed research builds upon two unrelated areas (optimization and stochastic dominance) and it is expected to promote a cross-fertilization of ideas that can potentially lead to further advances in both areas, while allowing for improved modeling abilities of application problems. This combination of different areas will also lead to the development of new graduate courses and the dissemination of ideas through a set of lecture notes on the topic.
