A single-phase, proximal path-following framework

Quoc Tran-Dinh, Anastasios Kyrillidis, Volkan Cevher

Published 2016-03-05Version 1

We propose a new proximal, path-following framework for a class of---possibly non-smooth---constrained convex problems. We consider settings where the non-smooth part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our main contribution is a new re-parametrization of the optimality condition of the barrier problem, that allows us to process the objective function with its proximal operator within a new path following scheme. In particular, our approach relies on the following two main ideas. First, we re-parameterize the optimality condition as an auxiliary problem, such that a "good" initial point is available. Second, we combine the proximal operator of the objective and path-following ideas to design a single phase, proximal, path-following algorithm. Our method has several advantages. First, it allows handling non-smooth objectives via proximal operators, this avoids lifting the problem dimension via slack variables and additional constraints. Second, it consists of only a \emph{single phase} as compared to a two-phase algorithm in [43] In this work, we show how to overcome this difficulty in the proximal setting and prove that our scheme has the same $\mathcal{O}(\sqrt{\nu}\log(1/\varepsilon))$ worst-case iteration-complexity with standard approaches [30, 33], but our method can handle nonsmooth objectives, where $\nu$ is the barrier parameter and $\varepsilon$ is a desired accuracy. Finally, our framework allows errors in the calculation of proximal-Newton search directions, without sacrificing the worst-case iteration complexity. We demonstrate the merits of our algorithm via three numerical examples, where proximal operators play a key role to improve the performance over off-the-shelf interior-point solvers.