Analysis of the Gradient Method with an Armijo-Wolfe Line Search on a Class of Nonsmooth Convex Functions

TL;DR

This paper analyzes the gradient method with Armijo-Wolfe line search on a class of nonsmooth convex functions, showing convergence to a specific point or unbounded descent depending on parameters, with experimental validation.

Contribution

It provides the first detailed analysis of the gradient method with inexact line search on certain nonsmooth convex functions, including convergence conditions and behavior.

Findings

Iterates converge to a point with zero first component when starting away from zero.

Under certain conditions, the function values tend to negative infinity.

Experimental results support the theoretical analysis.

Abstract

It has long been known that the gradient (steepest descent) method may fail on nonsmooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point. We give an analysis of the gradient method with steplengths satisfying the Armijo and Wolfe inexact line search conditions on the nonsmooth convex function $f(x) = a|x^{(1)}| + \sum_{i=2}^{n} x^{(i)}$. We show that if $a$ is sufficiently large, satisfying a condition that depends only on the Armijo parameter, then, when the method is initiated at any point $x_0 \in \R^n$ with $x^{(1)}_0\not = 0$, the iterates converge to a point $\bar x$ with $\bar x^{(1)}=0$, although $f$ is unbounded below. We also give conditions under which the iterates $f(x_k)\to-\infty$, using a specific Armijo-Wolfe bracketing line search. Our experimental results demonstrate that our analysis is reasonably tight.