A single-phase, proximal path-following framework

We propose a new proximal, path-following framework for a class of constrained convex problems. We consider settings where the nonlinear—and possibly non-smooth—objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. 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; by doing so, a family of alternative paths towards the optimum is generated. Second, we combine the proximal operator with 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 in order to accommodate nonsmooth components in optimization. Second, it consists of only a single phase: While the overall convergence rate of classical path-following schemes for self-concordant objectives does not suffer from the initialization phase, proximal path-following schemes undergo slow convergence, in order to obtain a good starting point [50]. In this work, we show how to overcome this limitation in the proximal setting and prove that our scheme has the same O( √ ν log(1/ε)) worst-case iteration-complexity with standard approaches [33, 37] without requiring an initial phase, where ν is the barrier parameter and ε is a desired accuracy. Finally, our framework allows errors in the calculation of proximal-Newton 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.