Comparison between MGDA and PAES for Multi-Objective Optimization

In multi-objective optimization, the knowledge of the Pareto set provides valuable information on the reachable optimal performance. A number of evolutionary strategies (PAES [4], NSGA-II [1], etc), have been proposed in the literature and proved to be successful to identify the Pareto set. However, these derivative-free algorithms are very demanding in terms of computational time. Today, in many areas of computational sciences, codes are developed that include the calculation of the gradient, cautiously validated and calibrated. Thus, an alternate method applicable when the gradients are known is introduced here. Using a clever combination of the gradients, a descent direction common to all criteria is identi ed. As a natural outcome, the Multiple Gradient Descent Algorithm (MGDA) is de ned as a generalization of steepest-descent method and compared with PAES by numerical experiments. Key-words: Optimization, gradient descent, Pareto optimality, Pareto front, performances in ria -0 06 05 42 3, v er si on 1 1 Ju l 2 01 1 Comparaison des algorithmes MGDA et PAES en optimisation multiobjectif Résumé : Dans le cadre d'une étude d'optimisation multiobjectif, la connaissance du front de Pareto permet de cerner e cacement le champ de recherche des paramètres optimaux. Pour ce faire, des algorithmes basés sur des méthodes évolutionnaires ont été développés (PAES [4], NSGA-II [1], etc). Nous proposons ici un algorithme alternatif, basé sur l'utilisation des gradients de critères permettant d'obtenir un échantillon du front de Pareto. Nous commençons par montrer qu'une combinaison judicieuse de ces gradients est une direction de descente commune à tous les critères. Mots-clés : Optimisation, gradient de descente, Pareto optimalitée, front de Pareto in ria -0 06 05 42 3, v er si on 1 1 Ju l 2 01 1 Testing Multiple-Gradient Descent Algorithm (MGDA) 3