  Perspective functions arise explicitly or implicitly in various forms in
applied mathematics and in statistical data analysis. To date, no systematic
strategy is available to solve the associated, typically nonsmooth,
optimization problems. In this paper, we fill this gap by showing that proximal
methods provide an efficient framework to model and solve problems involving
perspective functions. We study the construction of the proximity operator of a
perspective function under general assumptions and present important instances
in which the proximity operator can be computed explicitly or via
straightforward numerical operations. These results constitute central building
blocks in the design of proximal optimization algorithms. We showcase the
versatility of the framework by designing novel proximal algorithms for
state-of-the-art regression and variable selection schemes in high-dimensional
statistics.
