Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems

Abstract

The D-gap function approach has been adopted for solving variationalinequality problems. In this paper, we extend the approach forsolving equilibrium problems. From the theoretical point, we studythe convergence and global error bound of a D-gap function basedNewton method.   A general equilibrium problem is first formulated as an equivalentunconstrained minimization problem using a new D-gap function. Thenthe conditions of 'strict monotonicity' and 'strong monotonicity'for equilibrium problems are introduced. Under the strictmonotonicity condition, it is shown that a stationary point of theunconstrained minimization problem provides a solution to theoriginal equilibrium problem. Without the assumption of Lipschitzcontinuity, we further prove that strong monotonicity conditionguarantees the boundedness of the level sets of the new D-gapfunction and derive error bounds on the level sets. Combining thestrict monotonicity and strong monotonicity conditions, we show theexistence and uniqueness of a solution to the equilibrium problem,and establish the global convergence property of the proposedalgorithm with a global error bound.