A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms
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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 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.
KeywordsConvex and nonsmooth optimization Operator splitting Primal–dual algorithm Forward–backward method Douglas–Rachford method Monotone inclusion Proximal method Fenchel–Rockafellar duality
This work has been done at the author’s previous affiliation, the GREYC research center in Caen, France. The author wants to thank Jalal Fadili and Jean-Christophe Pesquet for stimulating discussions around splitting. He also thanks Isao Yamada for pointing out his article , which allowed to improve Lemma 4.3 in comparison with the result derived in a previous version of this manuscript. Moreover, the author is grateful to the anonymous referees and the associate editor for their valuable comments, which have contributed to the final preparation of the paper.
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