Optimization Online


Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDE's

H Attouch(attouch***at***math.univ-montp2.fr)
A Cabot(acabot***at***math.univ-montp2.fr)
P Frankel(p.frankel30***at***orange.fr)
J Peypouquet(juan.peypouquet***at***usm.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Citation: Unpublished, Université Montpellier II, Universidad Técnica Federico Santa María.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: 06/20/2010
Entry Accepted: 06/20/2010
Entry Last Modified: 06/20/2010

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Programming Society