The Proximal Gradient Method

Overview

ABSTRACT

The proximal gradient method is a splitting algorithm for the minimization of the sum of two convex functions, one of which is smooth. It has applications in areas such as mechanics, inverse problems, machine learning, image reconstruction, variational inequalities, statistics, operations research, and optimal transportation. Its formalism encompasses a wide variety of numerical methods in optimization such as gradient descent, projected gradient, iterative thresholding, alternating projections, the constrained Landweber method, as well as various algorithms in statistics and sparse data analysis. This paper aims at providing an account of the main properties of the proximal gradient method and to discuss some of its applications.

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AUTHOR

Patrick L. COMBETTES : North Carolina State University - Department of Mathematics - Raleigh, NC 27695, United States

INTRODUCTION

Notations. $H$ , $G$ and $G_{k}$ are Euclidean spaces, i.e. real Hilbertian spaces of finite dimension. Note $〈 \cdot | \cdot 〉$ their scalar product and $‖ \cdot ‖$ the associated norm. A function $f : H \to] – \infty, + \infty]$ is clean if $dom f = {x \in H | f (x) < + \infty} \neq \emptyset$ . The class of lower, convex and proper semicontinuous functions of $H$ in $] – \infty, + \infty]$ is noted $Γ_{0} (H)$ . Finally, $ir C$ ...

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KEYWORDS

splitting algorithm | convex function | numerical methods in optimization | gradient descent

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The Proximal Gradient Method

Mathematical framework

Bibliography

(1) - ACKER (F.), PRESTEL (M.A.) - Convergence d'un schéma de minimisation alternée, - Ann. Fac. Sci. Toulouse V. Sér. Math., vol. 2, pp. 1–9 (1980).
(2) - ATTOUCH (H.), BOLTE (J.), REDONT (P.), SOUBEYRAN (A.) - Alternating proximal algorithms for weakly coupled...

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