We study distributionally robust optimization (DRO) problems where the ambiguity set is de ned using the Wasserstein metric. We show that this class of DRO problems can be reformulated as semi-in nite programs. We give an exchange method to solve the reformulated problem for the general nonlinear model, and a central cutting-surface method for the convex case, … Read more
Scenarios are indispensable ingredients for the numerical solution of stochastic optimization problems. Earlier approaches for optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based on distances containing minimal information, i.e., on data appearing … Read more
We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of … Read more
Distributionally robust stochastic optimization (DRSO) is a framework for decision-making problems under certainty, which finds solutions that perform well for a chosen set of probability distributions. Many different approaches for specifying a set of distributions have been proposed. The choice matters, because it affects the results, and the relative performance of different choices depend on … Read more
The infinite models in integer programming can be described as the convex hull of some points or as the intersection of half-spaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for finite dimensional corner polyhedra. One consequence is that nonnegative continuous functions suffice to … Read more
Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set $K$. The idea consists of approximating from above the indicator function of $K$ with a sequence of polynomials of increasing degree $d$, so that the integrals of these polynomials generate a convergence sequence of upper bounds … Read more
We use moment methods to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate the infinite dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and … Read more
This paper proposes a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. … Read more
Quantifying the risk carried by an aggregate position $S_d\defn\sum_{i=1}^d X_i$ comprising many risk factors $X_i$ is fundamental to both insurance and financial risk management. Frechet inequalities quantify the worst-case risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate position in … Read more
Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is an underlying probability distribution that is known exactly, one hedges against a chosen set of distributions. In this paper, we consider sets of distributions that are within a chosen Wasserstein distance from a nominal distribution. … Read more