New concepts of directional derivatives for set-valued maps and applications to set optimization

ABSTRACT

We introduce and study two directional derivatives adapted to the case of set optimization. We motivate our approach by the novelty and the flexibility of these constructions for set optimization problems and by their relations, in some particular cases, to known objects of generalized differentiation. We prove optimality conditions for unconstrained problems on the basis of these derivatives and we extend these conditions to constrained problems by means of a penalization result and some calculus rules.

KEYWORDS:

• Set optimization
• directional derivatives
• optimality conditions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 54C60
• 46G05
• 90C46

Acknowledgements

The authors are thankful for the reviewers' constructive remarks that improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of the second and third authors was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS – UEFISCDI, project number PN-III-P4-PCE-2021-0690, within PNCDI III.