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

Azam Asl (aa2821***at***nyu.edu)
Michael L. Overton (mo1***at***nyu.edu )

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.

Keywords: Steepest descent method, Convex Optimization, Nonsmooth Optimization

Category 1: Convex and Nonsmooth Optimization

Citation: https://arxiv.org/abs/1711.08517

Download: [PDF]

Entry Submitted: 11/27/2017
Entry Accepted: 11/27/2017
Entry Last Modified: 09/29/2018

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