### Abstract

In this article, we consider an equilibrium problem: find a point uC such that f(u, y) 0 for all yC, where a continuous function [image omitted] satisfies f(x, x) = 0 for all [image omitted] and [image omitted] is a closed convex set. The existing computational methods for this problem require repetitive use of the metric projection onto C, which is often hard to compute. To relax the computational difficulty caused by the metric projection, we present a way to use any firmly nonexpansive mapping T satisfying [image omitted] in place of the metric projection onto C. The proposed method can be applied soundly to the Nash equilibrium problem in noncooperative games.

Original language | English |
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Pages (from-to) | 251-261 |

Number of pages | 11 |

Journal | Optimization |

Volume | 58 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Feb 2009 |

### Keywords

- Equilibrium problem
- Firmly nonexpansive mapping
- Metric projection
- Nash equilibrium problem
- Saddle point problem
- Subgradient-type method
- Variational inequality problem

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## Cite this

Iiduka, H., & Yamada, I. (2009). A subgradient-type method for the equilibrium problem over the fixed point set and its applications.

*Optimization*,*58*(2), 251-261. https://doi.org/10.1080/02331930701762829