Application of the Proximal Point Method to Nonmonotone Equilibrium Problems

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An inexact subgradient algorithm for Equilibrium Problems (2011, background): “In this work we assume that the function f (x, ∙) : Rn −→ (−∞, +∞] is convex and subdifferentiable at x , for all x ∈ C (see [8, 13, 15, 18]).”
Combined Relaxation Method for Mixed Equilibrium Problems (2005, background): “Due to (13), the sequence {y k } is bounded and there exists a bounded set W containing {x k } and {y k }. Using (8), ( 10 ), and (16), we have”
Extragradient algorithms extended to equilibrium problems (2008, methodology): “Moreover, if f is pseudomonotone on K, then K 1⁄4 K (see [14,21]).”
A hybrid extragradient method extended to fixed point problems and equilibrium problems (2013, background): “The equilibrium problems EP( f,C ) are very general in the sense that they include, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium (see [2,5,7,8,13]), .”
Equilibrium programming in Hilbert spaces (2005, background): “1) in this context in the case of a single bifunction (see [22, 23] and, in the case of Euclidean spaces, [13, 17, 20]), we are not aware of any result for systems of equilibrium problems.”
On the proximal point method for equilibrium problems in Hilbert spaces (2010, methodology): “Both [16] and [11] consider the regularized bifunction ~ f given by (2) and develop basically the same iterative procedure as we do here (see also [15] and [17]).”