  Convex nonsmooth optimization problems, whose solutions live in very high
dimensional spaces, have become ubiquitous. To solve them, the class of
first-order algorithms known as proximal splitting algorithms is particularly
adequate: they consist of simple operations, handling the terms in the
objective function separately. In this overview, we demystify a selection of
recent proximal splitting algorithms: we present them within a unified
framework, which consists in applying splitting methods for monotone inclusions
in primal-dual product spaces, with well-chosen metrics. Along the way, we
easily derive new variants of the algorithms and revisit existing convergence
results, extending the parameter ranges in several cases. In particular, we
emphasize that 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 for the convergence speed in practice.
