A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms

Laurent Condat (1)

1 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen

We propose a new first-order splitting algorithm for solving jointly the primal and dual formulations of large-scale convex minimization problems involving the sum of a smooth function with Lipschitzian gradient, a nonsmooth proximable function, and linear composite functions. This is a full splitting approach in the sense that the gradient and the linear operators involved are called explicitly without any inversion, while the nonsmooth functions are processed individually via their proximity operators. This work brings together and notably extends several classical splitting schemes like the forward-backward and Douglas-Rachford methods, as well as a recent primal-dual method designed by Chambolle and Pock for problems with linear composite terms.

Mots clés: Convex and nonsmooth optimization; operator splitting; primal-dual algorithm; forward-backward method; Douglas-Rachford method; monotone inclusion; proximal method; Fenchel-Rockafellar duality; saddle-point problem; monotone operator

Domaines: Optimisation et contrôle [math.OC]

Laurent Condat. A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. 2011. ⟨hal-00609728v3⟩