# American Institute of Mathematical Sciences

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

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 $$$$\left\{ \begin{array}{ll} x_{n+1}&=argmin\{f(\zeta) + \frac{1}{2\gamma}\|A\zeta - By_n\|^2_z +\frac{\alpha}{2}\|\zeta - x_n\|^2_x; \quad \zeta\in\mathcal{X}\}\\ y_{n+1}&=argmin\{g(\eta) + \frac{1}{2\gamma}\|Ax_{n+1} - B\eta\|^2_z +\frac{\nu}{2}\|\eta - y_n\|^2_y; \quad \eta\in\mathcal{Y}\}, \end{array} \right.$$$$ 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.
Citation: Pierre Frankel. Alternating proximal algorithm with costs-to-move, dual description and application to PDE's. Discrete and 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, Série Mathématiques, 2 (1980), 1-9. [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), 485-506. [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, Society for Industrial and Applied Mathematics (SIAM), Mathematical Programming Society (MPS), Philadelphia, PA, 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), 1061-1081. 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), 283-301. [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), 591-597. 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, Série Mathématiques, 2 (1980), 1-9. [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), 485-506. [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, Society for Industrial and Applied Mathematics (SIAM), Mathematical Programming Society (MPS), Philadelphia, PA, 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), 1061-1081. 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), 283-301. [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), 591-597. doi: 10.1090/S0002-9904-1967-11761-0.
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