For smooth objective functions it has been shown that the worst case cost of direct-search methods is of the same order as the one of steepest descent, when measured in number of iterations to achieve a certain threshold of stationarity. Motivated by the lack of such a result in the non-smooth case, we propose, analyze, … Read more
We investigate the BFGS algorithm with an inexact line search when applied to nonsmooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, assuming only absolute continuity. We also prove that the line search terminates … Read more
We investigate the behavior of the BFGS algorithm with an exact line search on nonsmooth functions. We show that it may fail on a simple polyhedral example, but that it apparently always succeeds on the Euclidean norm function, spiraling into the origin with a Q-linear rate of convergence; we prove this in the case of … Read more
Let $f$ be a continuous function on $\Rl^n$, and suppose $f$ is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which $f$ is not differentiable. Of particular interest is the case where $f$ is not convex, and perhaps not even locally Lipschitz, but … Read more