Augmented Lagrangian and Proximal Alternating Direction Methods of Multipliers in Hilbert spaces . Applications to Games , PDE ’ s and Control

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Augmented Lagrangian and Proximal Alternating Direction Methods of Multipliers in Hilbert spaces. Applications to Games, PDE’s and Control Hedy Attouch, Mohamed Soueycatt

[1]  P. Tallec Domain decomposition methods in computational mechanics , 1994 .

[2]  P. L. Combettes The foundations of set theoretic estimation , 1993 .

[3]  Paul-Emile Maingé,et al.  Strong convergence of an iterative method for hierarchical fixed point problems , 2007 .

[4]  Jonathan E. Spingarn,et al.  Applications of the method of partial inverses to convex programming: Decomposition , 1985, Math. Program..

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

[6]  R. Glowinski Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

[7]  S. A. Hirstoaga Approximation et résolution de problèmes d'équilibre, de point fixe et d'inclusion monotone , 2006 .

[8]  R. Glowinski,et al.  Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de Dirichlet non linéaires , 1975 .

[9]  Alexandre Cabot,et al.  Proximal Point Algorithm Controlled by a Slowly Vanishing Term: Applications to Hierarchical Minimization , 2005, SIAM J. Optim..

[10]  Paul Tseng,et al.  Alternating Projection-Proximal Methods for Convex Programming and Variational Inequalities , 1997, SIAM J. Optim..

[11]  Karl Kunisch,et al.  Semi-smooth Newton Methods for Non-differentiable Optimization Problems Lipschitz Lectures , 2008 .

[12]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[13]  Heinz H. Bauschke,et al.  Joint minimization with alternating Bregman proximity operators , 2005 .

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

[15]  F. Browder Nonlinear functional analysis , 1970 .

[16]  G. Alduncin On Gabay’s algorithms for mixed variational inequalities , 1997 .

[17]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[18]  Hai Yang,et al.  A Modified Variable-penalty Alternating Directions Method for Monotone Variational Inequalities , 2003 .

[19]  Heinz H. Bauschke,et al.  The asymptotic behavior of the composition of two resolvents , 2005, Nonlinear Analysis: Theory, Methods & Applications.

[20]  Kazufumi Ito,et al.  Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces , 2000 .

[21]  Jean-Pierre Dussault,et al.  Adaptive Scaling and Convergence Rates of a Separable Augmented Lagrangian Algorithm , 2000 .

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

[23]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[24]  Jonathan Eckstein Splitting methods for monotone operators with applications to parallel optimization , 1989 .

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

[26]  Masao Fukushima,et al.  Application of the alternating direction method of multipliers to separable convex programming problems , 1992, Comput. Optim. Appl..

[27]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[28]  A. Moudafi,et al.  Finding a Zero of The Sum of Two Maximal Monotone Operators , 1997 .

[29]  Z. Opial Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[30]  Bingsheng He,et al.  Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities , 1998, Oper. Res. Lett..

[31]  M. H. Xu Proximal Alternating Directions Method for Structured Variational Inequalities , 2007 .

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

[33]  Felipe Acker,et al.  Convergence d'un schéma de minimisation alternée , 1980 .

[34]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[35]  Marc Teboulle,et al.  A proximal-based decomposition method for convex minimization problems , 1994, Math. Program..

[36]  Patrick Redont,et al.  A New Class of Alternating Proximal Minimization Algorithms with Costs-to-Move , 2007, SIAM J. Optim..

[37]  RockaJellm MONOTONE OPERATORS ASSOCIATED WITH SADDLE . FUNCTIONS AND MINIMAX PROBLEMS R . 1 ' , 2022 .

[38]  Jacques Louis Lions,et al.  Contrôle des systèmes distribués singuliers , 1983 .

[39]  K. Kunisch,et al.  Primal-Dual Strategy for Constrained Optimal Control Problems , 1999 .

[40]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[41]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[42]  A. Moudafi Viscosity Approximation Methods for Fixed-Points Problems , 2000 .

[43]  Kazufumi Ito,et al.  An augmented Lagrangian technique for variational inequalities , 1990 .