April  2010, 6(2): 333-346. doi: 10.3934/jimo.2010.6.333

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

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084

2. 

Department of Mathematics, National Cheng Kung University, Tainan

3. 

Industrial Engineering and Operations Research, North Carolina State University, Raleigh, NC 27695-7906, United States

Received  March 2009 Revised  December 2009 Published  March 2010

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.
Citation: Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial and Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333
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