Multiple-Gradient Descent Algorithm (MGDA) for Pareto-Front Identification

Jean-Antoine Désidéri (1)

1 OPALE - Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE

This article compounds and extends several publications in which aMultiple-Gradient Descent Algorithm (MGDA), has been proposed and tested forthe treatment of multi-objective differentiable optimization. Originally introducedin [8], the method has been tested and reformulated in [9]. Its efficacy to identifythe Pareto front [4] has been demonstrated in [22], in comparison with an evolutionarystrategy. Recently, a variant, MGDA-II, has been proposed in which the descentdirection is calculated by a direct procedure [10] based on a Gram-Schmidtorthogonalization process (GSP) with special normalization. This algorithm wastested in the context of a simulation by domain partitioning, as a technique to matchthe different interface components concurrently [11]. The experimentation revealedthe importance of scaling, and a slightly modified normalization procedure wasproposed (”MGDA-IIb”). Two novel variants have been proposed since. The first,MGDA-III, realizes two enhancements. Firstly, the GSP is conducted incompletelywhenever a test reveals that the current estimate of the direction of search is adequatealso w.r.t. the gradients not yet taken into account; this improvement simplifiesthe identification of the search direction when the gradients point roughly in thesame direction, and makes the directional derivative common to several objectivefunctionslarger. Secondly, the order in which the different gradients are consideredin the GSP is defined in a unique way devised to favor an incomplete GSP. In thesecond variant, MGDA-IV, the question of scaling is addressed when the Hessiansare known. A variant is also proposed in which the Hessians are estimated by theBroyden-Fletcher-Goldfarb-Shanno (BFGS) formula. Lastly, a solution is proposedto adjust the step-size optimally in the descent step.

Mots clés: multi-objective optimization; descent direction; convex hull; BFGS quasi-Newton method; Gram-Schmidt orthogonalization process

Domaines: Mathématiques [math]; Sciences de l'ingénieur [physics]

Jean-Antoine Désidéri. Multiple-Gradient Descent Algorithm (MGDA) for Pareto-Front Identification. Fitzgibbon, W.; Kuznetsov, Y.A.; Neittaanmæki, P.; Pironneau, O. Modeling, Simulation and Optimization for Science and Technology, 34, Springer, 2014, Computational Methods in Applied Sciences, 978-94-017-9054-3. ⟨hal-01096049⟩