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Self-concordant inclusions: A unified framework for path-following generalized Newton-type algorithms by Tran-Dinh, Quoc; Sun, Tianxiao; Lu, Shu is a scholarly article available to read on EtoBox.

What is Self-concordant inclusions: A unified framework for path-following generalized Newton-type algorithms about?

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 $\mathcal{O}\left(\sqrt{\nu}\log(1/\varepsilon)\right)$-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.

Author

Tran-Dinh, Quoc; Sun, Tianxiao; Lu, Shu

Published

2017

Language

EN