A dynamical approach for the quantitative stability of parametric bilevel equilibrium problems and applications

Abstract

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results.

Keywords:

• Bilevel equilibrium problems
• equilibrium problems
• Fitzpatrick transform
• optimal control
• quantitative stability

AMS Subject Classifications:

• 90C33
• 90C31
• 49J40
• 46N10

Acknowledgments

The authors thank the anonymous referee for his valuable and constructive comments on the mathematical contribution of the initially submitted version and also for his nice suggestions concerning the presentation of the paper.

Disclosure statement

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