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

The D-gap function approach has been adopted for solving variational 
inequality problems. In this paper, we extend the approach for 
solving equilibrium problems. From the theoretical point, we study 
the convergence and global error bound of a D-gap function based 
Newton method. 
 
   A general equilibrium problem is first formulated as an equivalent 
unconstrained minimization problem using a new D-gap function. Then 
the conditions of "strict monotonicity" and "strong monotonicity" 
for equilibrium problems are introduced. Under the strict 
monotonicity condition, it is shown that a stationary point of the 
unconstrained minimization problem provides a solution to the 
original equilibrium problem. Without the assumption of Lipschitz 
continuity, we further prove that strong monotonicity condition 
guarantees the boundedness of the level sets of the new D-gap 
function and derive error bounds on the level sets. Combining the 
strict monotonicity and strong monotonicity conditions, we show the 
existence and uniqueness of a solution to the equilibrium problem, 
and establish the global convergence property of the proposed 
algorithm with a global error bound.