NDLI: Coupling Forward-Backward with Penalty Schemes and Parallel Splitting for Constrained Variational Inequalities

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Peypouquet, Juan Attouch, Hdy Czarnecki, Marc-Olivier
Copyright Year	2011
Abstract	We are concerned with the study of a class of forward-backward penalty schemes for solving variational inequalities $0\in Ax + N_C (x)$ where $\mathcal{H}$ is a real Hilbert space, $A: \mathcal{H}\rightrightarrows \mathcal{H}$ is a maximal monotone operator, and $N_C$ is the outward normal cone to a closed convex set $C\subset\mathcal{H}$. Let $\Psi: \mathcal{H} \to \mathbb R$ be a convex differentiable function whose gradient is Lipschitz continuous and which acts as a penalization function with respect to the constraint $x\in C.$ Given a sequence $(\beta_n)$ of penalization parameters which tends to infinity, and a sequence of positive time steps $(\lambda_n) \in\ell^2\setminus\ell^1$, we consider the diagonal forward-backward algorithm $x_{n+1}=(I+\lambda_nA)^{-1}(x_n-\lambda_n\beta_n \nabla \Psi (x_n)).$ Assuming that $(\beta_n)$ satisfies the growth condition $\limsup_{n\to\infty}\lambda_n\beta_n<2/\theta$ (where $\theta$ is the Lipschitz constant of $\nabla \Psi$), we obtain weak ergodic convergence of the sequence $(x_n)$ to an equilibrium for a general maximal monotone operator A. We also obtain weak convergence of the whole sequence $(x_n)$ when A is the subdifferential of a proper lower-semicontinuous convex function. As a key ingredient of our analysis, we use the cocoerciveness of the operator $\nabla \Psi$. When specializing our results to coupled systems, we bring new light to Passty's theorem and obtain convergence results of new parallel splitting algorithms for variational 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.
Starting Page	1251
Ending Page	1274
Page Count	24
File Format	PDF
ISSN	10526234
DOI	10.1137/110820300
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	4
Volume Number	21
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-11-01
Access Restriction	Subscribed
Subject Keyword	forward-backward algorithms Other methods constrained convex optimization Management decision making, including multiple objectives penalization methods maximal monotone operators parallel splitting methods variational inequalities Convex programming Dynamical systems in optimization and economics coupled systems Passty's theorem Optimization and variational techniques hierarchical optimization Mathematical programming methods Applications in optimization, convex analysis, mathematical programming, economics cocoercive operators compressive sensing
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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