Characterization of duality for a generalized quasi-equilibrium problem

Abstract

In this paper, the duality theory of a generalized quasi-equilibrium problem (also called generalized Ky Fan quasi-inequality) is investigated by using the image space approach. Generalized quasi-equilibrium problem is transformed into a minimization problem. The minimization problem is further reformulated as an image problem by virtue of linear/nonlinear separation function. The dual problem of the image problem is constructed in the image space, then zero duality gap between the image problem and its dual problem is derived under saddle point condition as well as the equivalent regular linear/nonlinear separation condition. Finally, some more sufficient conditions guaranteeing zero duality gap are also proposed.

Keywords:

• Generalized quasi-equilibrium problem (generalized Ky Fan quasi-inequality)
• zero duality gap
• image space
• saddle point
• regular separation

AMS Subject Classifications:

• 90C33
• 90C46

Acknowledgements

The authors are grateful to the referees and editors for their valuable comments and suggestions, which improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11171362], [grant number 11571055].