Some Dual Conditions for Global Weak Sharp Minimality of Nonconvex Functions

Abstract

Weak sharp minimality is a notion emerged in optimization whose utility is largely recognized in the convergence analysis of algorithms for solving extremum problems as well as in the study of the perturbation behavior of such problems. In this article, some dual constructions of nonsmooth analysis, mainly related to quasidifferential calculus and its recent developments, are employed in formulating sufficient conditions for global weak sharp minimality. They extend to nonconvex functions a condition, which is known to be valid in the convex case. A feature distinguishing the results here proposed is that they avoid to assume the Asplund property on the underlying space.

Keywords:

• Global weak sharp minimizers
• Lower exhausters
• Nondegeneracy condition
• Quasidifferential calculus
• Strong slope
• Variational principle

Mathematics Subject Classification:

• Primary: 49J52
• Secondary: 49J53, 90C48

ACKNOWLEDGMENTS

A. Uderzo would like to express his gratitude to the two anonymous referees for their constructive comments and helpful suggestions.

Part of the special issue, “Variational Analysis and Applications.”

Notes

¹For a statement of such principle the reader is referred to [19] (see Theorem 6.8.1). It is to be noted that, in the mentioned reference, being formulated in finite dimensional spaces, the principle is proved by using the proximinality property enjoyed by any nonempty closed subset of ℝ ⁿ . Nevertheless, with a slight modification, the proof can be rendered valid in any metric space.