On second-order Fritz John type optimality conditions for a class of differentiable optimization problems

Abstract

In this paper, we focus on the inequality constrained optimization problems, where the involved functions belong to the class $C^{1,1}$, i.e. functions which have Lipschitz continuous gradient mappings. We propose a second-order mean value inequality for a $C^{1,1}$ function in terms of its limiting second-order subdifferential. By virtue of this inequality, using the second-order tangent set and the asymptotic second-order tangent cone, we establish the second-order Fritz John type necessary and sufficient optimality conditions for differentiable optimization problems with inequality and set constraints.

COMMUNICATED BY:

• B. Mordukhovich

Keywords:

• Constrained optimization
• second-order optimality conditions
• second-order tangent sets
• limiting second-order subdifferentials

2010 Mathematical Reviews Classification Numbers:

• 49K30
• 49J53
• 90C46

Acknowledgments

We would like to thank an anonymous referee for helpful comments, which make the manuscript further refined.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was 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].