Alexander Shapiro’s research while affiliated with Georgia Institute of Technology and other places

In this paper, we consider asymptotics of the optimal value and the optimal solutions of parametric minimax estimation problems. Specifically, we consider estimators of the optimal value and the optimal solutions in a sample minimax problem that approximates the true population problem and study the limiting distributions of these estimators as the sample size tends to infinity. The main technical tool we employ in our analysis is the theory of sensitivity analysis of parameterized mathematical optimization problems. Our results go well beyond the existing literature and show that these limiting distributions are highly non-Gaussian in general and normal in simple specific cases. These results open up the way for the development of statistical inference methods in parametric minimax problems.

The main goal of this paper is to discuss the construction of distributionally robust counterparts of stochastic optimal control problems. Randomized and non-randomized policies are considered. In particular, necessary and sufficient conditions for the existence of non-randomized policies are given.

Optimization problems involving sequential decisions in a stochastic environment were studied in Stochastic Programming (SP), Stochastic Optimal Control (SOC) and Markov Decision Processes (MDP). This monograph concentrates on SP and SOC modeling approaches. In these frameworks, there are natural situations when the considered problems are convex. The classical approach to sequential optimization is based on dynamic programming. It has the problem of the so-called “curse of dimensionality”, in that its computational complexity increases exponentially with respect to the dimension of state variables. Recent progress in solving convex multistage stochastic problems is based on cutting plane approximations of the cost-to-go (value) functions of dynamic programming equations. Cutting plane type algorithms in dynamical settings is one of the main topics of this monograph. Also discussed in this work are stochastic approximation type methods applied to multistage stochastic optimization problems. From the computational complexity point of view, these two types of methods seem to be complimentary to each other. Cutting plane type methods can handle multistage problems with a large number of stages but a relatively smaller number of state (decision) variables. On the other hand, stochastic approximation type methods can only deal with a small number of stages but a large number of decision variables.

The main goal of this paper is to discuss several approaches to formulation of distributionally robust counterparts of Markov Decision Processes, where the transition kernels are not specified exactly but rather are assumed to be elements of the corresponding ambiguity sets. The intent is to clarify some connections between the game and static formulations of distributionally robust MDPs, and delineate the role of rectangularity associated with ambiguity sets in determining these connections.

This paper addresses decision making in multiple stages, where prior information is available and where consecutive and successive decisions are made. Risk measures assess the random outcome by taking various candidate probability measures into account. To justify decisions in multiple stages, it is essential to have conditional risk measures available, which respect the information, which was already revealed in the past. The paper addresses different variants of risk measures, discusses their properties in the specific context and their implications in multistage decision making. Various examples of risk measures on simple probability spaces with finite support illustrate the content. The Wasserstein and nested distance are involved to make decision making with numerous scenarios numerically tractalbe.

Optimization problems involving sequential decisions in a stochastic environment were studied in Stochastic Programming (SP), Stochastic Optimal Control (SOC) and Markov Decision Processes (MDP). In this paper we mainly concentrate on SP and SOC modelling approaches. In these frameworks there are natural situations when the considered problems are convex. Classical approach to sequential optimization is based on dynamic programming. It has the problem of the so-called ``Curse of Dimensionality", in that its computational complexity increases exponentially with increase of dimension of state variables. Recent progress in solving convex multistage stochastic problems is based on cutting planes approximations of the cost-to-go (value) functions of dynamic programming equations. Cutting planes type algorithms in dynamical settings is one of the main topics of this paper. We also discuss Stochastic Approximation type methods applied to multistage stochastic optimization problems. From the computational complexity point of view, these two types of methods seem to be complimentary to each other. Cutting plane type methods can handle multistage problems with a large number of stages, but a relatively smaller number of state (decision) variables. On the other hand, stochastic approximation type methods can only deal with a small number of stages, but a large number of decision variables.

The goal of this paper is to develop methodology for the systematic analysis of asymptotic statistical properties of data driven DRO formulations based on their corresponding non-DRO counterparts. We illustrate our approach in various settings, including both phi-divergence and Wasserstein uncertainty sets. Different types of asymptotic behaviors are obtained depending on the rate at which the uncertainty radius decreases to zero as a function of the sample size and the geometry of the uncertainty sets.