On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACT

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP.

KEYWORDS:

• Vector optimization
• Karush–Kuhn–Tucker conditions
• constraint qualifications
• second-order optimality conditions

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 26A24
• 49K99
• 90C29

Acknowledgements

We would like to thank an anonymous referee whose comments led to an improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Min Feng http://orcid.org/0000-0002-7439-0377

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant Numbers: 11571055, 11601437) and the Fundamental Research Funds for the Central Universities (Grant Number: 106112017CDJZRPY0020).