Auxiliary problem principles for equilibria. (2nd December 2017)

• Record Type: Journal Article
• Authors: Bigi, G.; Passacantando, M.
• Is Part Of: Optimization. Volume 66:Number 12(2017)
• Page Start: 1955
• Page End: 1972
• Publication Date: 2017-12-02
• Subjects: Equilibrium problem -- auxiliary problem -- Minty equilibrium problem -- gap function -- error bound; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1227808
• Languages: English
• ISSNs: 0233-1934

Abstract

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.