This result applies to both finite-dimensional and functional design spaces. This method converges to a point on the Pareto set. We propose to construct a fixed-point iteration in which updates of the element ω are used as successive directions of search. Applying this to the objective-function gradients (u_i = ∇J_i), we conclude that either ω = 0, and the current design point belongs to the Pareto set, or −ω is a descent direction common to all objective functions. Here, we propose a particular generalization of this method to multi-objective optimization by considering the concurrent minimization of n smooth criteria (u_i ∈ R^N, n ≤ N) in the convex hull of this family, there exists a unique element of minimal norm, say ω ∈ R^N then, the scalar product of ω with any vector in the convex hull, and in particular, with any u_i, is at least equal to ||ω||^2 ≥ 0. UCA - Université Côte d'Azur (Parc Valrose, 28, avenue Valrose 06108 Nice Cedex 2 - France) StructId : 1039632Ībstract : The steepest-descent method is a well-known and effective single-objective descent algorithm when the gradient of the objective function is known.CNRS - Centre National de la Recherche Scientifique : UMR7351 (France) StructId : 441569.COMUE UCA - COMUE Université Côte d'Azur (2015-2019) (Parc Valrose, 28, avenue Valrose 06108 Nice Cedex 2 - France) StructId : 523042.
UNS - Université Nice Sophia Antipolis (1965 - 2019) (Parc Valrose, 06100 Nice - France) StructId : 117617.Parc Valrose 06108 Nice Cedex 02 France - France) StructId : 199970 JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621 (Université Côte d'Azur U.M.R.Inria - Institut National de Recherche en Informatique et en Automatique (Domaine de Voluceauħ8153 Le Chesnay Cedex - France) StructId : 300009.CRISAM - Inria Sophia Antipolis - Méditerranée (2004 route des Lucioles BP 93 06902 Sophia Antipolis - France) StructId : 34586.1 OPALE - Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE (France) StructId : 2490
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