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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 |
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.
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