Forward-backward algorithm for functions with locally Lipschitz gradient: applications to mean field games

In this paper, we provide a generalization of the forward-backward splitting algorithm for minimizing the sum of a proper convex lower semicontinuous function and a differentiable convex function whose gradient satisfies a locally Lipschitztype condition. We prove the convergence of our method and derive a linear convergence rate when the differentiable function is locally strongly convex. We recover classical results in the case when the gradient of the differentiable function is globally Lipschitz continuous and an already known linear convergence rate when the function is globally strongly convex. We apply the algorithm to approximate equilibria of variational mean field game systems with local couplings. Compared with some benchmark algorithms to solve these problems, our numerical tests show similar performances in terms of the number of iterations but an important gain in the required computational time

Proximal method for geometry and texture image decomposition

International audienceWe propose a variational method for decomposing an image into a geometry and a texture component. Our model involves the sum of two functions promoting separately properties of each component, and of a coupling function modeling the interaction between the components. None of these functions is required to be differentiable, which significantly broadens the range of decompositions achievable through variational approaches. The convergence of the proposed proximal algorithm is guaranteed under suitable assumptions. Numerical examples are provided that show an application of the algorithm to image decomposition and restoration in the presence of Poisson noise

Proximal algorithms for multicomponent image recovery problems

International audienceIn recent years, proximal splitting algorithms have been applied to various monocomponent signal and image recovery problems. In this paper, we address the case of multicomponent problems. We first provide closed form expressions for several important multicomponent proximity operators and then derive extensions of existing proximal algorithms to the multicomponent setting. These results are applied to stereoscopic image recovery, multispectral image denoising, and image decomposition into texture and geometry components

Problèmes d'inclusions couplées : Éclatement, algorithmes et applications

This thesis is devoted to solving problems in set-valued nonlinear analysis in which several variables interact. The generic problem is modeled by an inclusion involving a sum of monotone operators in a product Hilbert space. Our objective is to design new algorithms for solving this problem under various sets of hypotheses on the underlying operators, and to study the asymptotic behavior of the resulting methods. A common property of the algorithms is the fact that they proceed by splitting in that the monotone operators and, if any, the linear operators present in the model act independently at each iteration. In particular, we address the case when the monotone operators are subdifferentials of convex functions, which leads to new minimization algorithms. The proposed methods unify and significantly extend the state-of-the art. They are applied to monotone inclusions in duality, to equilibrium problems, to signal and image processing, to game theory, to traffic theory, to evolution inclusions, to best approximation, and to domain decomposition in partial differential equations.Cette thèse est consacrée à la résolution de problèmes d'analyse non linéaire multivoque dans lesquels plusieurs variables interagissent. Le problème générique est modélisé par une inclusion vis-à-vis d'une somme d'opérateurs monotones sur un espace hilbertien produit. Notre objectif est de concevoir des nouveaux algorithmes pour résoudre ce problème sous divers jeux d'hypothèses sur les opérateurs impliqués et d'étudier le comportement asymptotique des méthodes élaborées. Une propriété commune aux algorithmes est le fait qu'ils procèdent par éclatement en ceci que les opérateurs monotones et, le cas échéant, les opérateurs linéaires constitutifs du modèle agissent indépendamment au sein de chaque itération. Nous abordons en particulier le cas où les opérateurs monotones sont des sous-différentiels de fonctions convexes, ce qui débouche sur de nouveaux algorithmes de minimisation. Les méthodes proposées unifient et dépassent largement l'état de l'art. Elles sont appliquées aux inclusions monotones composites en dualité, aux problèmes d'équilibre, au traitement du signal et de l'image, à la théorie des jeux, à la théorie du trafic, aux équations d'évolution, aux problèmes de meilleure approximation et à la décomposition de domaine dans les équations aux dérivées partielles

Forward-backward algorithm for functions with locally Lipschitz gradient: applications to mean field games

In this paper, we provide a generalization of the forward-backward splitting algorithm for minimizing the sum of a proper convex lower semicontinuous function and a differentiable convex function whose gradient satisfies a locally Lipschitztype condition. We prove the convergence of our method and derive a linear convergence rate when the differentiable function is locally strongly convex. We recover classical results in the case when the gradient of the differentiable function is globally Lipschitz continuous and an already known linear convergence rate when the function is globally strongly convex. We apply the algorithm to approximate equilibria of variational mean field game systems with local couplings. Compared with some benchmark algorithms to solve these problems, our numerical tests show similar performances in terms of the number of iterations but an important gain in the required computational time

Proximity Operators of Perspective Functions with Nonlinear Scaling

A perspective function is a construction which combines a base function defined on a given space with a nonlinear scaling function defined on another space and which yields a lower semicontinuous convex function on the product space. Since perspective functions are typically nonsmooth, their use in first-order algorithms necessitates the computation of their proximity operator. This paper establishes closed-form expressions for the proximity operator of a perspective function defined on a Hilbert space in terms of a proximity operator involving its base function and one involving its scaling function