An algorithmic approach to multiobjective optimization with decision uncertainty

In real life applications, optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one decision variable is a whole set, which includes all possible outcomes of this decision variable. We choose a robust approach and thus these sets have to be compared using the so-called upper-type less order relation. We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branch-and-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization, as convex underestimators, as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.