Proximal Splitting Algorithms: A Tour of Recent Advances, with New Twists.

Convex optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, proximal splitting algorithms are particularly adequate: they consist of simple operations, by handling the terms in the objective function separately. In this overview, we present a selection of recent proximal splitting algorithms within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metric. This allows us to derive new variants of the algorithms and to revisit existing convergence results, by extending the parameter ranges in several cases. In particular, when the smooth term in the objective function is quadratic, e.g. for least-squares problems, convergence is guaranteed with larger values of the relaxation parameter than previously known. Such larger values are usually beneficial to the convergence speed in practice.