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Borwein-Preiss Variational Principle Revisited

A. Y. Kruger (a.kruger***at***federation.edu.au)
S. Plubtieng (somyotp***at***nu.ac.th)
T. Seangwattana (seangwattana_t***at***hotmail.com)

Abstract: In this article, we refine and slightly strengthen the metric space version of the Borwein–Preiss variational principle due to Li, Shi, J. Math. Anal. Appl. 246, 308–319 (2000), clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein, Zhu, Techniques of Variational Analysis, Springer (2005) and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein, Preiss, Trans. Amer. Math. Soc. 303, 517-527 (1987) along several directions.

Keywords: Borwein-Preiss variational principle, smooth variational principle, gauge-type function, perturbation

Category 1: Convex and Nonsmooth Optimization

Citation: J. Math. Anal. Appl. 435 (2016) 1183–1193. http://dx.doi.org/10.1016/j.jmaa.2015.11.009

Download: [PDF]

Entry Submitted: 08/14/2015
Entry Accepted: 08/14/2015
Entry Last Modified: 12/15/2015

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