Active-set identification with complexity guarantees of an almost cyclic 2-coordinate descent method with Armijo line search

TL;DR

This paper proves that an almost cyclic 2-coordinate descent method with Armijo line search can identify active sets in finite steps for certain constrained problems, providing complexity guarantees under convexity.

Contribution

It introduces a novel analysis of active-set identification with complexity bounds for a simple coordinate descent method using Armijo line search.

Findings

Finite active-set identification for non-convex functions.

Complexity bounds under convexity and quadratic growth.

Method does not require exact minimizations.

Abstract

In this paper, it is established finite active-set identification of an almost cyclic 2-coordinate descent method for problems with one linear coupling constraint and simple bounds. First, general active-set identification results are stated for non-convex objective functions. Then, under convexity and a quadratic growth condition (satisfied by any strongly convex function), complexity results on the number of iterations required to identify the active set are given. In our analysis, a simple Armijo line search is used to compute the stepsize, thus not requiring exact minimizations or additional information.