On error bound moduli for locally Lipschitz and regular functions

• Published in: Mathematical programming Vol. 171; no. 1-2; pp. 463 - 487
• Main Authors: Li, M. H., Meng, K. W., Yang, X. Q.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2018; Springer Nature B.V
• Subjects: Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Constraining; Error analysis; Full Length Paper; Mathematical and Computational Physics; Mathematical functions; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Theoretical; Lower; 90C30; Locally Lipschitz; Outer limiting subdifferential; function; 90C34; Support function; End set; 65K10; Error bound modulus
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-017-1200-1

In this paper we study local error bound moduli for a locally Lipschitz and regular function via outer limiting subdifferential sets. We show that the distance from 0 to the outer limiting subdifferential of the support function of the subdifferential set, which is essentially the distance from 0 to the end set of the subdifferential set, is an upper estimate of the local error bound modulus. This upper estimate becomes tight for a convex function under some regularity conditions. We show that the distance from 0 to the outer limiting subdifferential set of a lower C 1 function is equal to the local error bound modulus.