Multiple-Gradient Descent Algorithm (MGDA)

نویسنده

  • Jean-Antoine Désidéri
چکیده

In a previous report [3], a methodology for the numerical treatment of a two-objective optimization problem, possibly subject to equality constraints, was proposed. The method was devised to be adapted to cases where an initial design-point is known and such that one of the two disciplines, considered to be preponderant, or fragile, and said to be the primary discipline, achieves a local or global optimum at this point. Then, a particular split of the design variables was proposed to accomplish a competitive-optimization phase by a Nash game, whose equilibrium point realizes an improvement of a secondary discipline, while causing the least possible degradation of the primary discipline from the initial optimum. In this new report, the initial design point and the number of disciplines are arbitrary. Certain theoretical results are established and they lead us to define a preliminary cooperative-optimization phase throughout which all the criteria improve, by a so-called Multiple-Gradient Descent Algorithm (MGDA), which generalizes to n disciplines (n ≥ 2) the classical steepest-descent method. This phase is conducted until a design-point on the Pareto set is reached; then, the optimization is interrupted or continued in a subsequent competitive phase by a generalization of the former approach by territory splitting and Nash game. Key-words: Optimum–shape design, concurrent engineering, multi-criterion optimization, split of territory, Nash and Stackelberg game strategies, Pareto optimality, descent direction, steepest-descent direction ∗ Project Team Opale: htpp://www-sop.inria.fr/opale ; [email protected] in ria -0 03 89 81 1, v er si on 2 5 Ju n 20 09 Algorithme de descente à gradients multiples (MGDA) Résumé : Dans un précédent rapport [3], on a proposé une méthodologie pour le traitement numérique d’un problème d’optimisation bicritère, éventuellement soumis à des contraintes d’égalité. La méthode a été conçue pour s’adapter aux cas où l’on connait un point de conception initial où l’une des disciplines, dite discipline principale car prépondérante, ou fragile, atteint un optimum local ou global. Alors, on a proposé une méthode de partage des variables afin de réaliser une phase d’optimisation compétitive par un jeu de Nash, dont le point d’équilibre produit une amélioration de la discipline secondaire, tout en causant la moindre dégradation possible de la discipline principale par rapport à l’optimum initial. Dans ce nouveau rapport, le point de conception initial et le nombre de disciplines sont arbitraires. On établit certains résultats théoriques qui nous conduisent à définir une phase préliminaire d’optimisation coopérative au cours de laquelle tous les critères s’améliorent, par unAlgorithme de Descente à Gradients Multiples (MGDA) qui généralise à n disciplines l’algorithme classique du gradient. Cette phase est prolongée jusqu’à atteindre un point de conception du front de Pareto; alors, on peut éventuellement continuer l’optimisation par une phase suivante, compétitive, en généralisant l’approche précédente par partage de territoire et jeu de Nash. Mots-clés : Conception optimale de forme, ingénierie concourante, optimisation multi-critère, partage de territoire, stratégies de jeux de Nash ou de Stackelberg, Pareto-optimalité, direction de descente, direction de plus grande pente in ria -0 03 89 81 1, v er si on 2 5 Ju n 20 09 Multiobjective optimization 3

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تاریخ انتشار 2009