Pareto front for two-stage distributionally robust optimization problems

Two-stage distributionally robust optimization is a recent optimization technique to handle uncertainty that is less conservative than robust optimization and more flexible than stochastic programming. The probability distribution of the uncertain parameters is not known but is assumed to belong to an ambiguity set. The size of certain types of ambiguity sets - such as several discrepancy-based ambiguity sets - is defined by a single parameter that makes it possible to control the degree of conservatism of the underlying optimization problem. Finding the values to assign to this parameter is a very relevant research topic. Hence, in this paper, we propose an exact and several heuristic methods for determining the control parameter values leading to all the relevant first-stage solutions. Our algorithmic approach resembles the $ϵ-$constrained method used to generate the Pareto front of a bi-objective problem. To demonstrate the applicability and efficacy of the proposed approaches, we conduct experiments on three different problems: scheduling, berth allocation, and facility location. The results obtained indicate that the proposed approaches provide sets of first-stage solutions very close to the optimal in a reasonable time.