Subdifferentials and derivatives with respect to a set and applications to optimization

ABSTRACT

In this article, we first examine some properties and calculus rules for the limiting subdifferential with respect to a set for functions. To do this, we propose corresponding derivatives with respect to a set for functions and study the relationships between these derivatives and the limiting subdifferential with respect to a set. We then find applications of these constructions in establishing optimality criteria for nonsmooth optimization problems. In particular, sufficient/necessary conditions for directionally optimal (strictly) solutions of both unconstrained and constrained nonsmooth optimization problems are provided and some of them are shown to be more useful in comparison with several existing results in the literature.

KEYWORDS:

• Subdifferential with respect to a set
• derivative
• optimality condition
• limiting subdifferential
• optimization problem

AMS SUBJECT CLASSIFICATIONS:

• 49K99
• 65K10
• 90C29
• 90C46

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) [grant number 101.01–2017.08].