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MGDA Variants for Multi-Objective Optimization

Jean-Antoine Désidéri 1
1 OPALE - Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : This report is a sequel to several publications in which a Multiple-Gradient Descent Algorithm (MGDA), has been proposed and tested for the treatment of multi-objective differentiable optimization. Originally introduced in [2], the method has been tested and reformulated in [6]. Its efficacy to identify the Pareto front has been demonstrated in [7], in comparison with an evolutionary strategy. Recently, a variant, MGDA-II, has been proposed in which the descent direction is calculated by a direct procedure [4] based on a Gram-Schmidt orthogonalization process (GSP) with special normalization. This algorithm was tested in the context of a simulation by domain partitioning, as a technique to match the different interface components concurrently [3]. The experimentation revealed the importance of scaling, and a slightly modified normalization procedure was proposed ("MGDA-IIb"). In this new report, two novel variants are proposed. The first, MGDA-III, realizes two enhancements. Firstly, the GSP is conducted incompletely whenever a test reveals that the current estimate of the direction of search is adequate also w.r.t. the gradients not yet taken into account; this improvement simplifies the identification of the search direction when the gradients point roughly in the same direction, and makes the Fréchet derivative common to several objective-functions larger. Secondly, the order in which the different gradients are considered in the GSP is defined in a unique way devised to favor an incomplete GSP. In the second variant, MGDA-IV, the question of scaling is addressed when the Hessians are known. A variant is also proposed in which the Hessians are estimated by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula.
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Jean-Antoine Désidéri. MGDA Variants for Multi-Objective Optimization. [Research Report] RR-8068, INRIA. 2012, pp.16. ⟨hal-00732881⟩

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