A Splitting Algorithm for Coupled System of Primal–Dual Monotone Inclusions

We propose a splitting algorithm for solving a coupled system of primal–dual monotone inclusions in real Hilbert spaces. The proposed algorithm has a structure identical to that of the forward–backward algorithm with variable metric. The operators involved in the problem formulation are used separately in the sense that single-valued operators are used individually and approximately in the forward steps and multi-valued operators are used individually via their generalization resolvent in the backward steps. The weak convergence of the algorithm proposed is proved. Applications to coupled system of monotone inclusions in duality and minimization problems, and multi-dictionary signal representation are demonstrated.

[1]  P. L. Combettes,et al.  Variable metric quasi-Fejér monotonicity , 2012, 1206.5705.

[2]  P. L. Combettes,et al.  Solving monotone inclusions via compositions of nonexpansive averaged operators , 2004 .

[3]  P. L. Combettes,et al.  Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators , 2011, Set-Valued and Variational Analysis.

[4]  Heinz H. Bauschke,et al.  Fixed-Point Algorithms for Inverse Problems in Science and Engineering , 2011, Springer Optimization and Its Applications.

[5]  Raymond H. Chan,et al.  Convergence analysis of tight framelet approach for missing data recovery , 2009, Adv. Comput. Math..

[6]  Patrick L. Combettes,et al.  Systems of Structured Monotone Inclusions: Duality, Algorithms, and Applications , 2012, SIAM J. Optim..

[7]  Patrick L. Combettes,et al.  A Parallel Splitting Method for Coupled Monotone Inclusions , 2009, SIAM J. Control. Optim..

[8]  Radu Ioan Bot,et al.  A Douglas-Rachford Type Primal-Dual Method for Solving Inclusions with Mixtures of Composite and Parallel-Sum Type Monotone Operators , 2012, SIAM J. Optim..

[9]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[10]  Patrick L. Combettes,et al.  Proximal Thresholding Algorithm for Minimization over Orthonormal Bases , 2007, SIAM J. Optim..

[11]  R. Tyrrell Rockafellar,et al.  Convergence Rates in Forward-Backward Splitting , 1997, SIAM J. Optim..

[12]  P. L. Combettes,et al.  Variable metric forward–backward splitting with applications to monotone inclusions in duality , 2012, 1206.6791.

[13]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[14]  Xiaojun Chen,et al.  A preconditioning proximal newton method for nondifferentiable convex optimization , 1997, Math. Program..

[15]  James V. Burke,et al.  On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating , 2000, Math. Program..

[16]  J. Pesquet,et al.  A Parallel Inertial Proximal Optimization Method , 2012 .

[17]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[18]  J. Baillon,et al.  Quelques propriétés des opérateurs angle-bornés etn-cycliquement monotones , 1977 .

[19]  Patrick L. Combettes,et al.  Proximal Algorithms for Multicomponent Image Recovery Problems , 2011, Journal of Mathematical Imaging and Vision.

[20]  U. Mosco.  Dual variational inequalities , 1972 .

[21]  Claude Lemaréchal,et al.  Variable metric bundle methods: From conceptual to implementable forms , 1997, Math. Program..

[22]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[23]  P. Tseng.  Applications of splitting algorithm to decomposition in convex programming and variational inequalities , 1991 .

[24]  Jian-Feng Cai,et al.  Simultaneous cartoon and texture inpainting , 2010 .

[25]  Benar Fux Svaiter,et al.  General Projective Splitting Methods for Sums of Maximal Monotone Operators , 2009, SIAM J. Control. Optim..

[26]  H. Attouch.  A General Duality Principle for the Sum of Two Operators 1 , 1996 .

[27]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[28]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[29]  P. L. Combettes,et al.  A proximal decomposition method for solving convex variational inverse problems , 2008, 0807.2617.

[30]  J. Bolte,et al.  Alternating Proximal Algorithms for Weakly Coupled Minimization Problems. Applications to Dynamical Games and PDE’s , 2008 .

[31]  J. Frédéric Bonnans,et al.  A family of variable metric proximal methods , 1995, Math. Program..

[32]  Heinz H. Bauschke,et al.  The Baillon-Haddad Theorem Revisited , 2009, 0906.0807.

[33]  Pablo A. Lotito,et al.  A Class of Inexact Variable Metric Proximal Point Algorithms , 2008, SIAM J. Optim..

[34]  P. L. Combettes,et al.  Dualization of Signal Recovery Problems , 2009, 0907.0436.

[35]  Mohamed-Jalal Fadili,et al.  A Generalized Forward-Backward Splitting , 2011, SIAM J. Imaging Sci..

[36]  Patrice Marcotte,et al.  Co-Coercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities , 1996, SIAM J. Optim..

[37]  P. L. Combettes,et al.  Monotone Operator Methods for Nash Equilibria in Non-potential Games , 2011, 1106.0144.

[38]  邵文革,et al.  Gilbert综合征二例 , 2009 .

[39]  R. Rockafellar,et al.  Duality and stability in extremum problems involving convex functions. , 1967 .

[40]  J. Burke,et al.  A Variable Metric Proximal Point Algorithm for Monotone Operators , 1999 .

[41]  Patrick L. Combettes,et al.  A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality , 2010, SIAM J. Optim..

[42]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[43]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[44]  B. Vu.  A Variable Metric Extension of the Forward–Backward–Forward Algorithm for Monotone Operators , 2012, 1210.2986.

[45]  Paul Tseng,et al.  Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming , 1990, Math. Program..

[46]  D. Gabay.  Applications of the method of multipliers to variational inequalities , 1983 .

[47]  Bang Công Vu,et al.  A splitting algorithm for dual monotone inclusions involving cocoercive operators , 2011, Advances in Computational Mathematics.

[48]  R. Boţ,et al.  Solving Systems of Monotone Inclusions via Primal-dual Splitting Techniques , 2013 .

[49]  B. Mercier.  Topics in Finite Element Solution of Elliptic Problems , 1979 .

[50]  Luis,et al.  Convex Variational Formulation with Smooth Coupling for Multicomponent Signal Decomposition and Recovery , 2009 .

[51]  Heinz H. Bauschke,et al.  Computational and Analytical Mathematics , 2013 .

[52]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .