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

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.

## Citation

Unpublished, Université Montpellier II, Universidad Técnica Federico Santa María.