The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primal-dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Comparisons with existing methods are made and applications to composite variational problems are demonstrated.2000 Mathematics Subject Classification: Primary 47H05; Secondary 65K05, 90C25.

A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces. Convergence is established for a wide class of coupling schemes. Unlike classical alternating algorithms, which are limited to two variables and linear coupling, our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions, dynamical games, signal recovery, image decomposition, best approximation, network flows, and variational problems in Sobolev spaces.where H 1 , H 2 , and G are Hilbert spaces, f 1 : H 1 → ]−∞, +∞] and f 2 : H 2 → ]−∞, +∞] are proper lower semicontinuous convex functions, and L 1 : H 1 → G and L 2 : H 2 → G are linear and bounded. This problem was solved in [5] via an inertial alternating minimization procedure first proposed in [9] for the case of the strongly coupled problem (1.2).The above problems and their solution algorithms are limited to two variables which, in addition, must be linearly coupled. These are serious restrictions since models featuring more than two variables and/or nonlinear coupling schemes arise naturally in applications. The purpose of this paper is to address simultaneously these restrictions by proposing a parallel algorithm for solving systems of monotone inclusions involving an arbitrary number of variables and nonlinear coupling. The breadth and flexibility of this framework will be illustrated through applications in the areas of evolution inclusions, dynamical games, signal recovery, image decomposition, best approximation, network flows, and decomposition methods in Sobolev spaces.We now state our problem formulation and our standing assumptions.

We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas-Rachford iteration involving the maximally monotone operator and the normal cone. In the second method it is a proximal step involving the partial inverse of the maximally monotone operator with respect to the vector subspace. Connections between the proposed methods and other methods in the literature are provided. Applications to monotone inclusions with finitely many maximally monotone operators and optimization problems are examined.2000 Mathematics Subject Classification: Primary 47H05; Secondary 47J25, 65K05, 90C25.

We address the numerical approximation of Mean Field Games with local couplings. For powerlike Hamiltonians, we consider both the stationary system introduced in [51,53] and also a similar system involving density constraints in order to model hard congestion effects [65,57]. For finite difference discretizations of the Mean Field Game system as in [3], we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results proved in [3]. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter, which can eventually be zero. We assess the performance of the methods via numerical experiments.

In 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.