Second-order optimality conditions for locally Lipschitz vector optimization problems

Abstract

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives.

Keywords:

• Locally Lipschitz
• second-order optimality conditions
• second-order Abadie constraint qualification
• strict local Pareto minimum of order 2

2020 AMS Classifications:

• 90C29
• 90C46

Disclosure statement

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

Additional information

Funding

The research of the first author is supported by CSIR-UGC, Junior Research Fellowship, India, National R&D Organisation (UGC-Ref. No. 1099/CSIR-UGC NET JUNE 2019) and the second author is supported by research grant under Faculty Research Programme of the IoE scheme of University of Delhi.