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Convex nonsmooth optimization problems, whose solutions live in very highdimensional spaces, have become ubiquitous. To solve them, the class offirst-order algorithms known as proximal splitting algorithms is particularlyadequate: they consist of simple operations, handling the terms in theobjective function separately. In this overview, we demystify a selection ofrecent proximal splitting algorithms: we present them within a unifiedframework, which consists in applying splitting methods for monotone inclusionsin primal-dual product spaces, with well-chosen metrics. Along the way, weeasily derive new variants of the algorithms and revisit existing convergenceresults, extending the parameter ranges in several cases. In particular, weemphasize that when the smooth term in the objective function is quadratic,e.g., for least-squares problems, convergence is guaranteed with larger valuesof the relaxation parameter than previously known. Such larger values areusually beneficial for the convergence speed in practice.

Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists