Nonsmooth Steepest Descent Method by Proximal Subdifferentials in Hilbert Spaces
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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.
KeywordsNonsmooth steepest descent method Stationary point Proximal subdifferential Prox-regularity
The authors are indebted to two anonymous referees for their helpful comments, which allowed us to improve the original presentation. This research was supported by the National Natural Science Foundation of P.R. China (Grant No. 11261067), the Scientific Research Foundation of Yunnan University under grant No. 2011YB29, and by IRTSTYN.
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