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All from the paper.
Much like gradient descent and the conjugate gradient method are standard tools of great use when optimizing smooth functions serially, proximal algorithms should be viewed as an analogous tool for nonsmooth, constrained, and distributed versions of these problems.
Proximal methods sit at a higher level of abstraction than classical optimization algorithms like Newton’s method. In such algorithms, the base operations are low-level, consisting of linear algebra operations and the computation of gradients and Hessians. In proximal algorithms, the base operations include solving small convex optimization problems (which in some cases can be done via a simple analytical formula).
Some problems are often more natural to solve using proximal algorithms rather than converting them to symmetric cone programs and using interior-point methods.