Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization

We study a generalization of the nonderivative discrete gradient method of Bagirov et al. for minimizing a locally Lipschitz function f on ℝn. We strengthen the existing convergence result for this method by showing that it either drives the f-values to −∞ or each of its cluster points is Clarke stationary for f, without requiring the compactness of the level sets of f. Our generalization is an approximate bundle method, which also subsumes the secant method of Bagirov et al.