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In this paper, we study the backward–forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward–backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward–backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient-projection algorithms, and give a numerical illustration of theoretical interest. [ABSTRACT FROM AUTHOR]

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators, whereis the subdifferential of a convex lower semicontinuous function, andis a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials which was considered in Abbas, Attouch, Svaiter JOTA, 2014. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems, thereby extending the known results. The time discretization of these dynamics gives various forward–backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multicriteria decision processes. [ABSTRACT FROM AUTHOR]

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving nonsmooth convex objective functions. Based on the Yosida regularization of the subdifferential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions, a key property for applications in cooperative games, inverse problems, and numerical multiobjective optimization. [ABSTRACT FROM AUTHOR]

We introduce a new class of forward-backward algorithms for structured convex minimization problems in Hubert spaces. Our approach relies on the time discretization of a second-order differential system with two potentials and Hessian-driven damping, recently introduced in [H. Attouch, P.-E. Maingé, and P. Redont, Differ. Equ. Appl, 4 (2012), pp. 27-65]. This system can be equivalently written as a first-order system in time and space, each of the two constitutive equations involving one (and only one) of the two potentials. Its time dicretization naturally leads to the introduction of forward-backward splitting algorithms with inertial features. Using a Liapunov analysis, we show the convergence of the algorithm under conditions enlarging the classical step size limitation. Then, we specialize our results to gradient-projection algorithms and give some illustrations of sparse signal recovery and feasibility problems. [ABSTRACT FROM AUTHOR]

We are concerned with the study of a class of forward-backward penalty schemes for solving variation inequalities 0 ∈ Ax + NC(x) where … is a real Hilbert space, A : H ⇉ H is a maximal monotone operator, and NC is the outward normal cone to a closed convex set C ⊂ H. Let Ψ : H → ℝ be a convex differentiable function whose gradient is Lipschitz continuous and which acts ,as a penalization function with respect to the constraint x ∈ C. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (λn) ∈ ℓ² \ ℓ¹, we consider the diagonal forward-backward algorithm xn+ 1 = (I + λnA)-1 (xn - λnβn∇Ψ(xn)). Assuming that (βn) satisfies the growth condition lim supn→∞ λnβn < 2/ϑ (where ϑ is the Lipschitz constant of ∇Ψ), we obtain weak ergodic convergence of the sequence (xn) to an equilibrium for a general maximal monotone operator A. We also obtain weak convergence of the whole sequence (xn) when A is the sub differential of a proper lower-semi continuous convex function. As a key ingredient of our analysis, we use the co coerciveness of the operator ∇Ψ. When specializing our results to coupled systems, we bring new light to Pasty's theorem and obtain convergence results of new parallel splitting algorithms for variation inequalities involving coupling in the constraint. We also establish robustness and stability results that account for numerical approximation errors. An illustration of compressive sensing is given. [ABSTRACT FROM AUTHOR]