Construction of ambiguity set in robust optimization relies on the choice of divergences between probability distributions. In distribution learning, choosing appropriate probability distributions based on observed data is critical for approximating the true distribution. To improve the performance of machine learning models, there has recently been interest in designing objective functions based on Lp-Wasserstein distance rather than the classical Kullback-Leibler (KL) divergence. In this paper, we derive concentration and asymptotic results using Bregman divergence. We propose a novel asymmetric statistical divergence called Wasserstein-Bregman divergence as a generalization of L2-Wasserstein distance. We discuss how these results can be applied to the construction of ambiguity set in robust optimization.
In @cite_22 , they formulate a robust optimization problem in terms of a KL divergence constraint and show that the problem can be converted into a convex optimization problem which can be solved analytically. In @cite_35 , they show that chance constraints with KL divergence ambiguity sets can be reformulated into a traditional chance constraint problem with different risk levels.
@cite_35 @cite_22
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In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general @math ź-divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general @math ź-divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various @math ź-divergence measures based on a capacitated lot-sizing problem with a quality-of-service requirement. In this paper we study distributionally robust optimization (DRO) problems where the ambiguity set of the probability distribution is defined by the Kullback-Leibler (KL) divergence. We consider DRO problems where the ambiguity is in the objective function, which takes a form of an expectation, and show that the resulted minimax DRO problems can be formulated as a one-layer convex minimization problem. We also consider DRO problems where the ambiguity is in the constraint. We show that ambiguous expectation-constrained programs may be reformulated as a one-layer convex optimization problem that takes the form of the Benstein approximation of Nemirovski and Shapiro (2006). We further consider distributionally robust probabilistic programs. We show that the optimal solution of a probability minimization problem is also optimal for the distributionally robust version of the same problem, and also show that the ambiguous chance-constrained programs (CCPs) may be reformulated as the original CCP with an adjusted confidence level. A number of examples and special cases are also discussed in the paper to show that the reformulated problems may take simple forms that can be solved easily. The main contribution of the paper is to show that the KL divergence constrained DRO problems are often of the same complexity as their original stochastic programming problems and, thus, KL divergence appears a good candidate in modeling distribution ambiguities in mathematical programming.