- Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDE's H Attouch(attouchmath.univ-montp2.fr) A Cabot(acabotmath.univ-montp2.fr) P Frankel(p.frankel30orange.fr) J Peypouquet(juan.peypouquetusm.cl) Abstract: Let $\cX,\cY,\cZ$ be real Hilbert spaces, let $f : \cX \rightarrow \R\cup\{+\infty\}$, $g : \cY \rightarrow \R\cup\{+\infty\}$ be closed convex functions and let $A : \cX \rightarrow \cZ$, $B : \cY \rightarrow \cZ$ be linear continuous operators. Let us consider the constrained minimization problem $$\min\{f(x)+g(y):\quad Ax=By\}.\leqno (\cP)$$ Given a sequence $(\gamma_n)$ which tends toward $0$ as $n\to+\infty$, we study the following alternating proximal algorithm \left\{ \begin{aligned} x_{n+1}&=\argmin\Big\{\gamma_{n+1}\,f(\zeta) + \frac{1}{2}\|A\zeta - By_n\|_\cZ^2 +\frac{\alpha}{2}\|\zeta - x_n\|_\cX^2; \,\,\, \zeta\in\cX\Big\}\\ y_{n+1}&=\argmin\Big\{\gamma_{n+1}\,g(\eta) + \frac{1}{2}\|Ax_{n+1} - B\eta\|_\cZ^2 +\frac{\nu}{2}\|\eta - y_n\|_\cY^2; \,\,\, \eta\in\cY\Big\}, \end{aligned} \right.\leqno (\cA) where $\alpha$ and $\nu$ are positive parameters. It is shown that if the sequence $\left({\gamma_n}\right)$ tends {\em moderately slowly} toward $0$, then the iterates of $(\cA)$ weakly converge toward a solution of $(\cP)$. The study is extended to the setting of maximal monotone operators, for which a general ergodic convergence result is obtained. Applications are given in the area of domain decomposition for PDE's. Keywords: Convex minimization, alternating minimization, proximal algorithm, variational inequalities, monotone inclusions, domain decomposition for PDE's Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Unpublished, Université Montpellier II, Universidad Técnica Federico Santa María. Download: [PDF]Entry Submitted: 06/20/2010Entry Accepted: 06/20/2010Entry Last Modified: 06/20/2010Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.