Variable Metric Forward-Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function

被引:135
作者
Chouzenoux, Emilie [1 ,2 ]
Pesquet, Jean-Christophe [1 ,2 ]
Repetti, Audrey [1 ,2 ]
机构
[1] Univ Paris Est Marne la Vallee, Lab Informat Gaspard Monge, F-77454 Champs Sur Marne, Marne La Vallee, France
[2] Univ Paris Est Marne la Vallee, CNRS, UMR 8049, F-77454 Champs Sur Marne, Marne La Vallee, France
关键词
Nonconvex optimization; Nonsmooth optimization; Majorize-Minimize algorithms; Forward-Backward algorithm; Image reconstruction; Proximity operator; SIGNAL; NOISE; OPTIMIZATION;
D O I
10.1007/s10957-013-0465-7
中图分类号
C93 [管理学]; O22 [运筹学];
学科分类号
070105 ; 12 ; 1201 ; 1202 ; 120202 ;
摘要
We consider the minimization of a function G defined on , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka-Aojasiewicz property. Such a problem can be solved with the Forward-Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize-Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward-Backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.
引用
收藏
页码:107 / 132
页数:26
相关论文
共 50 条
[1]  
[Anonymous], 2006, GRUNDLEHREN MATH WIS
[2]  
[Anonymous], 1999, Athena scientific Belmont
[3]  
[Anonymous], 1963, Les quations aux Drives Partielles, DOI DOI 10.1006/JDEQ.1997.3393
[4]  
[Anonymous], 1999, SPRINGER SCI
[5]  
[Anonymous], 1998, GRUND MATH WISS
[6]  
[Anonymous], 1996, Die Grundlehren der mathematischen Wissenschaften
[7]  
[Anonymous], 2007, GRADIENT METHODS MIN
[8]   On the convergence of the proximal algorithm for nonsmooth functions involving analytic features [J].
Attouch, Hedy ;
Bolte, Jerome .
MATHEMATICAL PROGRAMMING, 2009, 116 (1-2) :5-16
[9]   Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods [J].
Attouch, Hedy ;
Bolte, Jerome ;
Svaiter, Benar Fux .
MATHEMATICAL PROGRAMMING, 2013, 137 (1-2) :91-129
[10]   Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems [J].
Beck, Amir ;
Teboulle, Marc .
IEEE TRANSACTIONS ON IMAGE PROCESSING, 2009, 18 (11) :2419-2434