Analysis of the gradient method with an Armijo–Wolfe line search on a class of non-smooth convex functions

It has long been known that the gradient (steepest descent) method may fail on non-smooth 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 non-smooth convex function (Formula presented.). 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 (Formula presented.) with (Formula presented.), the iterates converge to a point (Formula presented.) with (Formula presented.), although f is unbounded below. We also give conditions under which the iterates (Formula presented.), using a specific Armijo–Wolfe bracketing line search. Our experimental results demonstrate that our analysis is reasonably tight.