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Alternating proximal algorithm with costs-to-move, dual description and application to PDE's

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  • Given real Hilbert spaces $\mathcal{X},\mathcal{Y},\mathcal{Z}$, closed convex functions $f : \mathcal{X} \longrightarrow \mathbb{R}\cup\{+\infty\}$, $g : \mathcal{Y} \longrightarrow \mathbb{R}\cup\{+\infty\}$ and linear continuous operators $A : \mathcal{X} \longrightarrow \mathcal{Z}$, $B : \mathcal{Y} \longrightarrow \mathcal{Z}$, we study the following alternating proximal algorithm $$ \begin{equation} \left\{ \begin{array}{ll} x_{n+1}&=argmin\{f(\zeta) + \frac{1}{2\gamma}\|A\zeta - By_n\|^2_z +\frac{\alpha}{2}\|\zeta - x_n\|^2_x; \quad \zeta\in\mathcal{X}\}\\ y_{n+1}&=argmin\{g(\eta) + \frac{1}{2\gamma}\|Ax_{n+1} - B\eta\|^2_z +\frac{\nu}{2}\|\eta - y_n\|^2_y; \quad \eta\in\mathcal{Y}\}, \end{array} \right. \end{equation} $$ where $\gamma$, $\alpha$ and $\nu$ are positive parameters. Under suitable conditions, we prove that any sequence $(x_n,y_n)$ generated by $(\mathcal{A})$ weakly converges toward a minimum point of the function $(x,y)\mapsto f(x) + g(y) + \frac{1}{2\gamma}\|Ax - By\|^2_z$ and that the sequence of dual variables $\left(-\frac{1}{\gamma}(Ax_n-By_n)\right)$ strongly converges in $\mathcal{Z}$ toward the unique minimizer of the function $z\mapsto f^*(A^*z)+g^*(-B^*z)+\frac{\gamma}{2}\|z\|^2_z$. An application is given in variational problems and PDE's.
    Mathematics Subject Classification: 65K05, 65K10, 49J40, 90C25.


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