Auxiliary problem principles for equilibria

Abstract

The auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a given EP with each of these auxiliary problems are investigated exploiting parametric definitions of different kinds of convexity and monotonicity. This analysis leads to extending some known results for variational inequalities and linear EPs to the general case together with new equivalences. Stationarity and convexity properties of gap functions are investigated as well in this framework. Moreover, both new results on the existence of a unique solution and new error bounds based on gap functions with good convexity properties are obtained under weak quasimonotonicity or weak concavity assumptions.

Keywords:

• Equilibrium problem
• auxiliary problem
• Minty equilibrium problem
• gap function
• error bound

Acknowledgements

The authors wish to thank the two anonymous referees and the associate editor for the careful reviews and valuable comments, which helped improve the paper. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA – National Group for Mathematical Analysis, Probability and their Applications) of the Istituto Nazionale di Alta Matematica (INdAM – National Institute of Higher Mathematics).

Notes

No potential conflict of interest was reported by the authors.