Wasserstein distributionally robust chance-constrained program with moment information

Abstract

This paper studies a distributionally robust joint chance-constrained program with a hybrid ambiguity set including the Wasserstein metric, and moment and bounded support information of uncertain parameters. For the considered mathematical program, the random variables are located in a given support space, so a set of random constraints with a high threshold probability for all the distributions that are within a specified Wasserstein distance from an empirical distribution, and a series of moment constraints have to be simultaneously satisfied. We first demonstrate how to transform the distributionally robust joint chance-constrained program into an equivalent reformulation, and show that such a program with binary variables can be equivalently reformulated as a mixed 0–1 integer conic program. To reduce the computational complexity, we derive a relaxed approximation of the joint DRCCP-H using McCormick envelop relaxation, and introduce linear relaxed and conservative approximations by using norm-based inequalities when the Wasserstein metric uses the lp-norm with p≠1 and p≠∞. Finally, we apply this new scheme to address the multi-dimensional knapsack and surgery block allocation problems. The results show that the model with a hybrid ambiguity set yields less conservative solutions when encountering uncertainty over the model with an ambiguity set involving only the Wasserstein metric or moment information, verifying the merit of considering the hybrid ambiguity set, and that the linear approximations significantly reduce the computational time while maintaining high solution quality.