Gap finding and validation in evolutionary multi- and many-objective optimization

Over 30 years, evolutionary multi- and many-objective optimization (EMO/EMaO) algorithms have been extensively applied to find well-distributed Pareto-optimal (PO) solutions in a single run. However, in real-world problems, the PO front may not always be a single continuous hyper-surface, rather several irregularities may exist involving disjointed surfaces, holes within the surface, or patches of mixed-dimensional surfaces. When a set of trade-off solutions are obtained by EMO/EMaO algorithms, there may exist less dense or no solutions (we refer as 'gaps') in certain parts of the front. This can happen for at least two reasons: (i) gaps naturally exist in the PO front, or (ii) no natural gaps exists, but the chosen EMO/EMaO algorithm is not able to find any solution in the apparent gaps. To make a confident judgement, we propose a three-step procedure here. First, we suggest a computational procedure to identify gaps, if any, in the EMO/EMaO-obtained PO front. Second, we propose a computational method to identify well-distributed gap-points in the gap regions. Third, we apply a focused EMO/EMaO algorithm to search for possible representative trade-off points in the gaps. We then propose two metrics to qualitatively establish whether a gap truly exists in the obtained dataset, and if yes, whether the gap naturally exists on the true Pareto-set. Procedures are supported by results on two to five-objective test problems and on a five-objective scheduling problem from a steel-making industry.