Computation of the improvement directions of the Pareto front and its application to MOEAs

This paper introduces the mathematical development and algorithm of the Improvement-Directions Mapping (IDM) method, which computes improvement directions to "push" the current solutions toward the true Pareto front. The main idea is to compute normal vectors to the front, as improvement directions in the objective space, to be then transformed into search directions in the variable space through a transformation tensor. The main contributions of the IDM as a local search operator versus previous approaches are the following: 1) It does not require of a priori information about improvement directions or location of the true Pareto front, 2) It uses a local quadratic approximation of the Pareto front to compute the transformation tensor, thus, reducing numerical problems and avoiding abrupt changes in the search direction which could lead to erratic searches. These features allow the IDM to be implemented as a local search operator within any Multi-objective Evolutionary Algorithm (MOEA). The potential of the IDM is shown by hybridizing two well-known multi-objective algorithms: a) MOEA/D + IDM; b) NSGA-II + IDM. In the first approach, IDM "pushes" the offspring population in each iteration. A similar experiment is performed with the second approach. Furthermore, one more experiment evaluates the IDM as a refinement step that is applied to the last Pareto front delivered by NSGA-II.