Nonsmooth Steepest Descent Method by Proximal Subdifferentials in Hilbert Spaces

In this paper, we first study a nonsmooth steepest descent method for nonsmooth functions defined on a Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox‐regularity and Lipschitz continuity. As an application, the established algorithm is used to search for the minimizer of a lower semicontinuous and convex function on a finite‐dimensional space. A convergence theorem, as an extension and improvement of the existing converging result for twice continuously differentiable convex functions, is also presented therein.