# Self-concordant inclusions: A unified framework for path-following generalized Newton-type algorithms

arXiv: https://arxiv.org/abs/1707.07403 · HTML: https://mlsanity.com/paper/1707.07403/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2017.

## Built on by (most cited first, 2 of 2)

- [Sublinear Convergence Rates of Extragradient-Type Methods: A Survey on Classical and Recent Developments](https://mlsanity.com/paper/2303.17192/) (2023)
- [Evaluating and Learning Robust Bandit Policies Under Uncertain Causal Mechanisms](https://mlsanity.com/paper/2508.02812/) (2025)

## Abstract (arXiv)

We study a class of monotone inclusions called "self-concordant inclusion" which covers three fundamental convex optimization formulations as special cases. We develop a new generalized Newton-type framework to solve this inclusion. Our framework subsumes three schemes: full-step, damped-step and path-following methods as specific instances, while allows one to use inexact computation to form generalized Newton directions. We prove a local quadratic convergence of both the full-step and damped-step algorithms. Then, we propose a new two-phase inexact path-following scheme for solving this monotone inclusion which possesses an Oleft(√nulog(1/varepsilon)right)-worst-case iteration-complexity to achieve an varepsilon-solution, where nu is the barrier parameter and varepsilon is a desired accuracy. As byproducts, we customize our scheme to solve three convex problems: convex-concave saddle-point, nonsmooth constrained convex program, and nonsmooth convex program with linear constraints. We also provide three numerical examples to illustrate our theory and compare with existing methods.