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arXiv · 2503.14479

La Méthode du Gradient Proximé

Patrick L. Combettes
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Abstract

The proximal gradient method is a splitting algorithm for the minimization of the sum of two convex functions, one of which is smooth. It has applications in areas such as mechanics, inverse problems, machine learning, image reconstruction, variational inequalities, statistics, operations research, and optimal transportation. Its formalism encompasses a wide variety of numerical methods in optimization such as gradient descent, projected gradient, iterative thresholding, alternating projections, the constrained Landweber method, as well as various algorithms in statistics and sparse data analysis. This paper aims at providing an account of the main properties of the proximal gradient method and to discuss some of its applications. --- La méthode du gradient proximé est un algorithme d'éclatement pour la minimisation de la somme de deux fonctions convexes, dont l'une est lisse. Elle trouve des applications dans des domaines tels que la mécanique, le traitement du signal, les problèmes inverses, l'apprentissage automatique, la reconstruction d'images, les inéquations variationnelles, les statistiques, la recherche opérationnelle et le transport optimal. Son formalisme englobe une grande variété de méthodes numériques en optimisation, telles que la descente de gradient, le gradient projeté, la méthode de seuillage itératif, la méthode des projections alternées, la méthode de Landweber contrainte, ainsi que divers algorithmes en statistique et en analyse parcimonieuse de données. Cette synthèse vise à donner un aperçu des principales propriétés de la méthode du gradient proximé et d'aborder certaines de ses applications.

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Patrick L. Combettes. 2025-05-13. La Méthode du Gradient Proximé. https://arxiv.org/abs/2503.14479

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