Statistics

Regular polygon

Optimization problem

Programming language

Divergence (linguistics)

Ambiguity

Kullback–Leibler divergence

Stochastic programming

Computer science

Mathematical optimization

Probability distribution

Probabilistic logic

Mathematics

Minimax

Robust optimization

Convex optimization

Linguistics

Philosophy

Geometry

Constraint (computer-aided design)

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 c...

Kullback-Leibler divergence constrained distributionally robust optimization