A dominance-based stability measure for multi-objective evolutionary algorithms

Over the years, we have been applying multi- objective evolutionary algorithms (MOEAs) to a number of real- world problems, solving multi-objective optimization problems (MOPs) in the real world faces a number of challenges including when to terminate the algorithm. This paper addresses this challenge by introducing what we call a "stability measure". We use this measure to estimate when to stop the multi-objective evolutionary search. For the proposed measure, the non-dominated set obtained by an MOEA will be tested under local variability in the decision variable space. A non-dominated solution found by the MOEA will be assigned a stability value, which corresponds to the number of solutions within the neighborhood that dominate it. Obviously, if the found non-dominated solution lies on the Pareto optimal front (POF) then there cannot be any such dominating solutions in its neighborhood. The average of stability values assigned to all non-dominated solutions will be used as the stability value for the set. In order to validate the proposed measure, we carried out measurements on the obtained non-dominated sets from two MOEAs (NSGA-II and SPEA2), and two other well-known algorithms hill-climber, and a random-walk. We use random- walk in order to create a baseline for judging the performance of the algorithms, where we expect the highest level of variations to occur. To apply this measure at each generation, it incurs additional cost that does not contribute to the evolutionary search per se. This motivated us to add this measure as a local search operator. In this way, the local search operator plays two roles: (1) it attempts to find better solutions than the ones we have in the population; and (2) it acts as a stability measure for the evolutionary search. The results confirmed the usefulness of the proposed measure and algorithm.