- Analysis of the Gradient Method with an Armijo-Wolfe Line Search on a Class of Nonsmooth Convex Functions Azam Asl (aa2821nyu.edu) Michael L. Overton (mo1nyu.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: 11/27/2017Entry Accepted: 11/27/2017Entry Last Modified: 09/29/2018Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.