Search

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]