# A single-phase, proximal path-following framework

arXiv: https://arxiv.org/abs/1603.01681 · HTML: https://mlsanity.com/paper/1603.01681/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2016.

## Built on by (most cited first, 1 of 1)

- [Momentum-inspired Low-Rank Coordinate Descent for Diagonally Constrained SDPs](https://mlsanity.com/paper/2106.08775/) (2021)

## Abstract (arXiv)

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 non-smooth 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 citeTranDinh2013e. In this work, we show how to overcome this limitation in the proximal setting and prove that our scheme has the same O(√nulog(1/varepsilon)) worst-case iteration-complexity with standard approaches citeNesterov2004,Nesterov1994 without requiring an initial phase, where nu is the barrier parameter and varepsilon 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.