To cite this version: , vol. 158, no. 2, pp. 460-479, 2013. AbstractWe 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 applied 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 the recent primal-dual method of Chambolle and Pock designed for problems with linear composite terms.

A new algorithm is proposed to project, exactly and in finite time, a vector of arbitrary size onto a simplex or an ℓ 1-norm ball. It can be viewed as a Gauss-Seidel-like variant of Michelot's variable fixing algorithm; that is, the threshold used to fix the variables is updated after each element is read, instead of waiting for a full reading pass over the list of non-fixed elements. This algorithm is empirically demonstrated to be faster than existing methods.