Parameter Selection and Pre-Conditioning for a Graph Form Solver

In a recent paper, Parikh and Boyd describe a method for solving a convex optimization problem, where each iteration involves evaluating a proximal operator and projection onto a subspace. In this paper we address the critical practical issues of how to select the proximal parameter in each iteration, and how to scale the original problem variables, so as the achieve reliable practical performance. The resulting method has been implemented as an open-source software package called POGS (Proximal Graph Solver), that targets multi-core and GPU-based systems, and has been tested on a wide variety of practical problems. Numerical results show that POGS can solve very large problems (with, say, more than a billion coefficients in the data), to modest accuracy in a few tens of seconds. As just one example, a radiation treatment planning problem with around 100 million coefficients in the data can be solved in a few seconds, as compared to around one hour with an interior-point method.