On the use of Grossone Methodology for Handling Priorities in Multi-Objective Evolutionary Optimization

This chapter introduces a new class of optimization problems, called Mixed Pareto-Lexicographic Multi-objective Optimization Problems (MPL-MOPs), to provide a suitable model for scenarios where some objectives have priority over some others. Specifically, this work focuses on two relevant subclasses of MPL-MOPs, namely optimization problems having the objective functions organized as emph{priority chains} or emph{priority levels}. A priority chain (PC) is a sequence of objectives ordered lexicographically by importance; conversely, a priority level (PL) is a group of objectives having the same importance in terms of optimization, but a lexicographic ordering exists between the PLs. After describing these problems and discussing why the standard algorithms are inadequate, an innovative approach to deal with them is introduced: it leverages the Grossone Methodology, a recent theory that allows handling priorities by means of infinite and infinitesimal numbers. Most interestingly, this technique can be easily embedded in most of the existing evolutionary algorithms, without altering their core logic. Three algorithms for MPL-MOPs are shown: the first two, called PC-NSGA-II and PC-MOEA/D, are the generalization of NSGA-II and MOEA/D, respectively, in the presence of PCs; the third, named PL-NSGA-II, generalizes instead NSGA-II when PLs are present. Several benchmark problems, including some from the real world, are used to evaluate the effectiveness of the proposed approach. The generalized algorithms are compared to other famous evolutionary ones, either priority-based or not, through a statistical analysis of their performances. The experiments show that the generalized algorithms are consistently able to produce more solutions and of higher quality.