Distributionally Robust Stochastic Optimization with Wasserstein Distance

12/1/16 | 4:15pm | E51-315 
Reception to follow.

Abstract:  Consider an optimization problem under uncertainty. One may consider formulating it as a stochastic optimization problem. Often in such a setting, a "true" probability distribution may not be known. In fact, often the notion of a true probability distribution may not even be applicable. We consider an approach, called distributionally robust stochastic optimization (DRSO), in which one hedges against all probability distributions in a chosen set. We point out that the popular sets based on phi-divergences such as Kullback-Leibler divergence have poor properties for some problems, and that sets based on Wasserstein distance have more appealing properties. Motivated by that observation we consider distributionally robust stochastic optimization problems that hedge against all probability distributions that are within a chosen Wasserstein distance from a nominal distribution, for example an empirical distribution resulting from available data. Such a choice of sets has two advantages: (1) The resulting distributions hedged against are more reasonable than those resulting from sets based on phi-divergences. (2) The problem of determining the worst-case expectation over the resulting set of distributions has desirable tractability properties. We derive a dual reformulation of the corresponding DRSO problem and construct worst-case distributions explicitly via the first-order optimality conditions of the dual problem.

Our contributions are five-fold.

(i) We identify necessary and sufficient conditions for the existence of a worst-case distribution, which are naturally related to the growth rate of the objective function.
(ii) We show that the worst-case distributions resulting from an appropriate Wasserstein distance have a concise structure and a clear interpretation.
(iii) Using this structure, we show that data-driven DRSO problems can be approximated to any accuracy by robust optimization problems, and thereby many DRSO problems become tractable by using tools from robust optimization.
(iv) To the best of our knowledge, our proof of strong duality is the first constructive proof for DRSO problems, and we show that the constructive proof technique is also useful in other contexts.
(v) Our strong duality result holds in a very general setting, and we show that it can be applied to infinite dimensional process control problems and worst-case value-at-risk analysis.

This is joint work with Rui Gao at Georgia Tech.

Anton Kleywegt

Associate Professor
Georgia Tech

Bio: Anton Kleywegt is an associate professor in the Stewart School of Industrial & Systems Engineering at Georgia Tech. Dr. Kleywegt conducts research in optimization and stochastic modeling with applications in transportation, distribution, and logistics, especially in the following areas: vehicle routing and scheduling, inventory routing, distribution operations, fleet assignment, vendor managed inventory, distribution network design, yield management, terminal design and operations, logistics planning and control, multi-modal transportation, and intelligent transportation systems. He has also worked with Praxair, Columbian Chemicals Company, Delta Air Lines, Manhattan Associates, and The Home Depot on SCL projects in addressing logistics research in vendor managed inventory, fleet sizing and allocation, revenue management, scheduling of order picking, and distribution planning. Dr. Kleywegt received a Ph.D. from the School of Industrial Engineering at Purdue University in 1996 and joined the ISyE faculty this same year as an assistant professor. 

Event Time: 

2016 - 16:15