### Embedded Evolutionary Multi-Objective Optimization for Worst Case Robustness

Abstract

In Multi-Objective Problems (MOPs) involving uncertainty, each solution might be associated with a cluster of performances in the objective space depending on the possible scenarios. Therefore, in MOPs, the worst case might not be a single scenario but rather a set of such worst case scenarios, depending on the user preferences. The evolution of solutions based on their related sets of worst case scenarios has been recently introduced. It has been termed: "worst case evolutionary multi-objective optimization." In the current paper the worst case evolutionary multi-objective optimization is further developed. In contrast to the former work where the number of possible scenarios is small and the set of worst cases can thus be easily determined, here, the number of scenarios is assumed to be large, and the worst cases are searched for by means of an embedded evolutionary search. This means that for each nominal solution, a worst set of scenarios has to be found. In the current study, the resulting front, consisting of sets of solutions' worst cases, is formally defined, and a new approach to support decision making based on it, is suggested. The new decision support poses the selection as an auxiliary MOP, highlighting the tradeoff which might result from the worst being a set and not a single point. An academic example and an engineering design problem are given in order to explain the methodology and to demonstrate its applicability to real life problems.