### Abstract

In this paper, we introduce a new approach for solving equilibrium problems in Hilbert spaces. First, we transform the equilibrium problem into the problem of finding a zero of a sum of two maximal monotone operators. Then, we solve the resulting problem using the Glowinski–Le Tallec splitting method and we obtain a linear rate of convergence depending on two parameters. In particular, we enlarge significantly the range of these parameters given rise to the convergence. We prove that the sequence generated by the new method converges to a global solution of the considered equilibrium problem. Finally, numerical tests are displayed to show the efficiency of the new approach.

Original language | English |
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Pages (from-to) | 477–495 |

Number of pages | 19 |

Journal | Journal of Global Optimization |

Volume | 70 |

Issue number | 2 |

DOIs | |

Publication status | Published - 9 Oct 2017 |

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### Keywords

- Equilibrium problem
- Global convergence
- Glowinski–Le Tallec splitting method
- Maximal monotone operator
- Nash equilibrium

### Cite this

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*Journal of Global Optimization*, vol. 70, no. 2, pp. 477–495. https://doi.org/10.1007/s10898-017-0575-0

**The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces.** / Vuong, Phan Tu; Strodiot, Jean Jacques.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces

AU - Vuong, Phan Tu

AU - Strodiot, Jean Jacques

PY - 2017/10/9

Y1 - 2017/10/9

N2 - In this paper, we introduce a new approach for solving equilibrium problems in Hilbert spaces. First, we transform the equilibrium problem into the problem of finding a zero of a sum of two maximal monotone operators. Then, we solve the resulting problem using the Glowinski–Le Tallec splitting method and we obtain a linear rate of convergence depending on two parameters. In particular, we enlarge significantly the range of these parameters given rise to the convergence. We prove that the sequence generated by the new method converges to a global solution of the considered equilibrium problem. Finally, numerical tests are displayed to show the efficiency of the new approach.

AB - In this paper, we introduce a new approach for solving equilibrium problems in Hilbert spaces. First, we transform the equilibrium problem into the problem of finding a zero of a sum of two maximal monotone operators. Then, we solve the resulting problem using the Glowinski–Le Tallec splitting method and we obtain a linear rate of convergence depending on two parameters. In particular, we enlarge significantly the range of these parameters given rise to the convergence. We prove that the sequence generated by the new method converges to a global solution of the considered equilibrium problem. Finally, numerical tests are displayed to show the efficiency of the new approach.

KW - Equilibrium problem

KW - Global convergence

KW - Glowinski–Le Tallec splitting method

KW - Maximal monotone operator

KW - Nash equilibrium

UR - http://www.mendeley.com/research/glowinskile-tallec-splitting-method-revisited-framework-equilibrium-problems-hilbert-spaces

U2 - 10.1007/s10898-017-0575-0

DO - 10.1007/s10898-017-0575-0

M3 - Article

VL - 70

SP - 477

EP - 495

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

IS - 2

ER -