Auxiliary problem principles for equilibria

Giancarlo Bigi and Mauro Passacantando


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 which depend upon a regularization parameter α: the classical auxiliary problems (CAEPs) and the Minty EPs (MEPs).
The goal of the paper is to analyse in detail the conditions that guarantee the equivalence between EP and each of the above auxiliary problems together with the properties and advantages that each equivalence brings. A rather novel feature is the analysis of CAEPs for negative α's which, up to now, has been considered only for the so-called linear EP. Furthermore, a systematic analysis of MEPs allows achieving new equivalence results also for positive α's.
Section 2 explores the connections between the convexity and monotonicity properties of the bifunction f of EP and its regularization. Section 3 and Section 4 investigate the relationships of EP with CAEPs and MEPs, respectively, and the properties of the corresponding gap functions, which allow reformulating EP as optimization programs. Exploiting suitable values for α, also new results on the existence of a unique solution, stationarity properties of gap functions and error bounds are achieved under weak monotonicity or weak concavity assumptions on the bifunction f.

The original paper is available at 10.1080/02331934.2016.1227808.

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