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

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June 2012, 5(3): 545-557. doi: 10.3934/dcdss.2012.5.545

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

1.	Département de Mathématiques, Université Montpellier II, CC 051, Place Eugène, Bataillon, 34095 Montpellier Cedex 5, France

Received April 2010 Revised May 2010 Published October 2011

Abstract
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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.

Keywords: Convex minimization, proximal algorithm, alternating minimization, domain decomposition for PDE's..

Mathematics Subject Classification: 65K05, 65K10, 49J40, 90C2.

Citation: Pierre Frankel. Alternating proximal algorithm with costs-to-move, dual description and application to PDE's. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 545-557. doi: 10.3934/dcdss.2012.5.545

References:

[1]	F. Acker and M.-A. Prestel, Convergence d'un schéma de minimisation alternée,, Annales de la Faculté des Sciences de Toulouse V, 2 (1980), 1.
[2]	H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's,, Journal of Convex Analysis, 15 (2008), 485.
[3]	H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces. Applications to PDE's and Optimization,", MPS/SIAM Series on Optimization, 6 (2006).
[4]	H. Attouch, A. Cabot, P. Frankel and J. Peypouquet, Alternating proximal algorithms for constrained variational inequalities. Applications to domain decomposition for PDE's,, accepted in Nonlinear Analysis., ().
[5]	H. Attouch, P. Redont and A. Soubeyran, A new class of alternating proximal minimization algorithms with costs-to-move,, SIAM Journal on Optimization, 18 (2007), 1061. doi: 10.1137/060657248.
[6]	H. H. Bauschke, P. L. Combettes and S. Reich, The asymptotic behavior of the composition of two resolvents,, Nonlinear Analysis, 60 (2005), 283.
[7]	A. Cabot and P. Frankel, Alternating proximal algorithms with costs-to-move and asymptotically vanishing coupling,, accepted in Optimization., ().
[8]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 591. doi: 10.1090/S0002-9904-1967-11761-0.

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References:

[1]	F. Acker and M.-A. Prestel, Convergence d'un schéma de minimisation alternée,, Annales de la Faculté des Sciences de Toulouse V, 2 (1980), 1.
[2]	H. Attouch, J. Bolte, P. Redont and A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's,, Journal of Convex Analysis, 15 (2008), 485.
[3]	H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces. Applications to PDE's and Optimization,", MPS/SIAM Series on Optimization, 6 (2006).
[4]	H. Attouch, A. Cabot, P. Frankel and J. Peypouquet, Alternating proximal algorithms for constrained variational inequalities. Applications to domain decomposition for PDE's,, accepted in Nonlinear Analysis., ().
[5]	H. Attouch, P. Redont and A. Soubeyran, A new class of alternating proximal minimization algorithms with costs-to-move,, SIAM Journal on Optimization, 18 (2007), 1061. doi: 10.1137/060657248.
[6]	H. H. Bauschke, P. L. Combettes and S. Reich, The asymptotic behavior of the composition of two resolvents,, Nonlinear Analysis, 60 (2005), 283.
[7]	A. Cabot and P. Frankel, Alternating proximal algorithms with costs-to-move and asymptotically vanishing coupling,, accepted in Optimization., ().
[8]	Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 591. doi: 10.1090/S0002-9904-1967-11761-0.

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