On Well-Posedness for Perturbed Quasi-Equilibrium and Quasi-Optimization Problems

Abstract

We propose definitions of well-posedness under perturbation in terms of suitable asymptotically solving sequences, not embedding a given problem into a parameterized family. When considering in detail quasi-equilibrium and quasi-optimization problems as well as optimization problems with equilibrium constraints, we establish sufficient conditions for both well-posedness and unique well-posedness. For cases with noncompact underlying sets of constraints, measures of noncompactness are employed. We provide numerous examples to ensure the essentialness of the assumptions imposed in the obtained results.

Keywords:

• Asymptotically solving sequence
• measure of noncompactness
• quasi-equilibrium problem
• quasi-optimization problem
• quasi-optimization problem with equilibrium constraints
• (unique) well-posedness

MATHEMATICS SUBJECT CLASSIFICATION:

• 49K40
• 90C31
• 65K99

Acknowledgment

The authors are very grateful to the Editors and Referees for their valuable remarks and suggestions that helped us significantly improve the paper. For the first and the last authors, this is a result of the project under Grant number B2021-TCT-02 supported by the Ministry of Education and Training of Viet Nam.

Additional information

Funding

The first and the last authors are supported by the project under grant number B2021-TCT-02 supported by Ministry of Education and Training of Viet Nam, the second author is supported by the Foundation for Science and Technology Development (NAFOSTED) under the grant number 101.01-2020.23.