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

Phan Tu Vuong, Jean Jacques Strodiot

Research output: Contribution to journalArticle

### 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 477–495 19 Journal of Global Optimization 70 2 https://doi.org/10.1007/s10898-017-0575-0 Published - 9 Oct 2017

### Fingerprint

Splitting Method
Hilbert spaces
Equilibrium Problem
Hilbert space
Maximal Monotone Operator
Global Solution
Two Parameters
Rate of Convergence
Transform
Converge
Zero
Range of data
Framework
Equilibrium problem

### Keywords

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

### Cite this

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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.",
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author = "Vuong, {Phan Tu} and Strodiot, {Jean Jacques}",
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day = "9",
doi = "10.1007/s10898-017-0575-0",
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In: Journal of Global Optimization, Vol. 70, No. 2, 09.10.2017, p. 477–495.

Research output: Contribution to journalArticle

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

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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

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