Some Methods for Solving Quasi-Equilibrium Problems

Let X be a given nonempty closed convex subset in IR and let f : X×X −→ IR be a given bifunction with f(x, x) = 0 for all x ∈ X. We are interested in the following quasi-equilibrium problem { Find x ∈ K(x) such that f(x, y) ≥ 0 ∀ y ∈ K(x) (QEP ) where K(·) is a multivalued mapping X −→ 2 . When the mapping x −→ K(x) is such that K(x) = X for all x ∈ X, (QEP ) reduces naturally to the equilibrium problem: { Find x ∈ X such that f(x, y) ≥ 0 ∀ y ∈ X (EP ) In the case when f(x, y) = F (x) (y − x) in (QEP ) with F : X −→ IR, we obtain an important class of problems, namely the class of quasi-variational inequality problems: { Find x ∈ K(x) such that F (x) (y − x) ≥ 0 ∀ y ∈ K(x) (QV IP ) Note that the generalized Nash equilibrium problem (GNEP ), which is an important model fruitfully used in many different applications, can be reformulated as a (QEP ) or (QV IP ). The (EP ), (GNEP ), (QV IP ) and (V IP ) have received considerable attention in recent years from theoretical viewpoint, applications and solution methodologies. Compared with the (QEP ), the literature on the algorithms for the quasi-equilibrium problems is not as extensive. See, for example, the following recent surveys and the references cited therein: − M. Pappalardo, G. Mastroeni, M. Passacantando, Merit functions: a bridge between optimization and equilibria, 4OR A Quartely Journal of Operations Research 12 (2014) 1–33. − G. Bigi, M. Castellani, M. Pappalardo, M. Passacantando, Existence and solution methods for equilibria, European Journal of Operational Research 227 (2013) 1–11. − F. Facchinei, C. Kanzow, Generalized Nash equilibrium problems, Annals of Operations Research 175 (2010) 177–211.