Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Attouch, H., Cominetti, R.: A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differ. Equ. 128, 519–540 (1996)
Bazaraa, M.S., Shetty, C.M.: Nonlinear Programming: Theory and Algorithm. Wiley, New York (1979)
Chuong, T.D., Yao, J.-C.: Steepest descent methods for critical points in vector optimization problems. Optimization 91, 1811–1829 (2012)
Grana Drummond, L.M., Svaiter, B.F.: A steepest descent method for vector optimization. J. Comput. Appl. Math. 175, 395–414 (2005)
Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51, 479–494 (2000)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)
Polak, E.: Computational Methods in Optimization. Academic Press, New York (1971)
Smyrlis, G., Zisis, V.: Local convergence of the steepest descent method in Hilbert spaces. J. Math. Anal. Appl. 300, 436–453 (2004)
Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)
Fletcher, R., Reeves, C.M.: Functions minimization by conjugate gradients. Comput. J. 7, 149–154 (1964)
Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, Berlin (1998)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I/II. Springer, Berlin (2006)
Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Poliquin, R., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Am. Math. Soc. 348(5), 1805–1838 (1996)
Bernard, F., Thilbaut, L.: Uniform prox-regularity of functions and epigraphs in Hilbert spaces. Nonlinear Anal. 60, 187–207 (2005)
Bacak, M., Borwein, J.M., Eberhard, A., Mordukhovich, B.S.: Infimal convolutions and Lipschitzian properties of subdifferentials for prox-regular functions in Hilbert spaces. J. Convex Anal. 17, 737–763 (2010)
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Math., vol. 1364. Springer, New York (1989)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Boris S. Mordukhovich.
Rights and permissions
About this article
Cite this article
Wei, Z., He, Q.H. Nonsmooth Steepest Descent Method by Proximal Subdifferentials in Hilbert Spaces. J Optim Theory Appl 161, 465–477 (2014). https://doi.org/10.1007/s10957-013-0444-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0444-z