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

Abstract

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 the local quadratic convergence of both 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 $${\mathcal {O}}(\sqrt{\nu }\log (1/\varepsilon ))$$-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: the convex–concave saddle-point problem, the nonsmooth constrained convex program, and the nonsmooth convex program with linear constraints. We also provide three numerical examples to illustrate our theory and compare with existing methods.