On variational inequalities using directional convexificators

ABSTRACT

In this paper, we give some results which constitute an application of directional convexificators recently introduced by Dempe and Pilecka [Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming. J Global Optim. 2015;61:769–788]. After establishing mean value conditions in terms of directional convexificators, we formulate variational inequalities of Stampacchia and Minty type in terms of directional convexificators and use these variational inequalities as a tool to find out necessary and sufficient conditions for a point to be an optimal solution of an inherent optimization problem. An example illustrating our findings is also given.

Keywords:

• Directional convexificators
• continuity directions
• nonsmooth optimization
• variational inequalities
• ${\mathrm{\partial }}_D^{\ast }$-convexity

Acknowledgments

The authors sincere acknowledgements to the anonymous referees for their insightful remarks and suggestions. Dedicated to Riahi Hassan in honor of his 60th birthday, with great respect.

Disclosure statement

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