Primal-Dual Proximal Algorithms for Structured Convex Optimization: A Unifying Framework

We present a simple primal-dual framework for solving structured convex optimization problems involving the sum of a Lipschitz-differentiable function and two nonsmooth proximable functions, one of which is composed with a linear mapping. The framework is based on the recently proposed asymmetric forward-backward-adjoint three-term splitting (AFBA); depending on the value of two parameters, (extensions of) known algorithms as well as many new primal-dual schemes are obtained. This allows for a unified analysis that, among other things, establishes linear convergence under four different regularity assumptions for the cost functions. Most notably, linear convergence is established for the class of problems with piecewise linear-quadratic cost functions.