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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

Pierre Frankel 1,

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 $$ \begin{equation} \left\{ \begin{array}{ll} x_{n+1}&=argmin\{f(\zeta) + \frac{1}{2\gamma}\|A\zeta - By_n\|^2_z +\frac{\alpha}{2}\|\zeta - x_n\|^2_x; \quad \zeta\in\mathcal{X}\}\\ y_{n+1}&=argmin\{g(\eta) + \frac{1}{2\gamma}\|Ax_{n+1} - B\eta\|^2_z +\frac{\nu}{2}\|\eta - y_n\|^2_y; \quad \eta\in\mathcal{Y}\}, \end{array} \right. \end{equation} $$ 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, Série Mathématiques, 2 (1980), 1-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Cabot and P. Frankel, Alternating proximal algorithms with costs-to-move and asymptotically vanishing coupling,, accepted in Optimization., ().   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Cabot and P. Frankel, Alternating proximal algorithms with costs-to-move and asymptotically vanishing coupling,, accepted in Optimization., ().   Google Scholar

[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.  Google Scholar

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