Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.

[1]  W. Fenchel.  Convex cones, sets, and functions , 1953 .

[2]  L. Sandgren.  On convex cones , 1954 .

[3]  R. Rockafellar.  Minimax Theorems and Conjugate Saddle-Functions. , 1964 .

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

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

[6]  Lynn McLinden.  An extension of Fenchel’s duality theorem to saddle functions and dual minimax problems , 1974 .

[7]  R. Temam,et al.  Analyse convexe et problèmes variationnels , 1974 .

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

[9]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

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

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

[12]  B. Mercier.  Inequations variationnelles de la mécanique , 1980 .

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

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

[15]  Decision Systems.,et al.  Convolutions of maximal monotone mappings , 1986 .

[16]  Gregory B. Passty.  The parallel sum of nonlinear monotone operators , 1986 .

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

[18]  R. Rockafellar.  Conjugate Duality and Optimization , 1987 .

[19]  Alberto Seeger,et al.  Direct and inverse addition in convex analysis and applications , 1990 .

[20]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .

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

[22]  Abdellatif Moudafi.  On the stability of the parallel sum of maximal monotone operators , 1996 .

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

[24]  Masao Fukushima,et al.  The primal douglas-rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem , 1996, Math. Program..

[25]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[26]  Stephen M. Robinson,et al.  Composition duality and maximal monotonicity , 1999, Math. Program..

[27]  Michael C. Ferris,et al.  Smooth methods of multipliers for complementarity problems , 1999, Math. Program..

[28]  Teemu Pennanen,et al.  Dualization of Generalized Equations of Maximal Monotone Type , 1999, SIAM J. Optim..

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

[30]  Xiaoqi Yang,et al.  Duality in optimization and variational inequalities , 2002 .

[31]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[32]  I. V. Konnov.  The splitting method with linear search for primal-dual variational inequalities , 2003 .

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

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

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

[36]  Antonin Chambolle,et al.  Dual Norms and Image Decomposition Models , 2005, International Journal of Computer Vision.

[37]  Heinz H. Bauschke,et al.  A new proximal point iteration that converges weakly but not in norm , 2005 .

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

[39]  Gonzalo Alduncin,et al.  Composition duality principles for mixed variational inequalities , 2005, Math. Comput. Model..

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

[41]  P. L. Combettes,et al.  Proximity for sums of composite functions , 2010, 1007.3535.

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

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

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

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