Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

ABSTRACT

Second-order optimality conditions for the vector nonlinear programming problems with inequality constraints and continuously differentiable data are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce notions of a second-order pseudoconvex vector function and KKT-pseudoconvex vector problem with inequality constraints. We derive necessary and sufficient conditions for efficiency in the vector problem with inequality constraints. Three examples are presented.

KEYWORDS:

• Multiobjective nonsmooth optimization
• Karush-Kuhn-Tucker optimality conditions
• characterizations of the solution set
• second-order pseudoconvex function
• second-order Mangasarian-Fromovitz constraint qualifications

AMS CLASSIFICATIONS:

• 90C46
• 90C26
• 90C29
• 26B25
• 49J52

Acknowledgements

The author would like to express his gratitude to the anonymous referees for their helpful comments on the first version of the paper. This research is partially supported by the TU-Varna Grant No 19/2019.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Vsevolod I. Ivanov http://orcid.org/0000-0001-7685-4908

Additional information

Funding

This research is partially supported by the TU-Varna Grant No 19/2019.