On Robust Optimization

Abstract

We introduce an unconstrained multicriteria optimization problem and discuss its relation to various well-known scalar robust optimization problems with a finite uncertainty set. Specifically, we show that a unique solution of a robust optimization problem is Pareto optimal for the unconstrained optimization problem. Furthermore, it is demonstrated that the set of weakly Pareto optimal solutions of the unconstrained multicriteria optimization problem contains all solutions of certain scalar robust optimization problems. An example is presented to verify our results. In addition, we show that the set of solutions of a weighted robust optimization problem always contains Pareto optimal solutions of the unconstrained multicriteria optimization problem. Similarly, we indicate that the set of solutions of a strictly robust optimization problem comprises Pareto optimal points of the unconstrained vector-valued problem. By assembling all these results we point out strong relations between unconstrained vector optimization and the more intuitively introduced concepts of scalar robust optimization. Finally, we provide a sufficient condition for an optimal solution of a strictly robust optimization problem.