A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces

Research output: Contribution to journalArticle

Abstract

A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported.

Original languageEnglish
Pages (from-to)159-178
Number of pages20
JournalJournal of Global Optimization
Volume64
Issue number1
DOIs
Publication statusPublished - 1 Jan 2016

Fingerprint

Extragradient Method
Hilbert spaces
Equilibrium Problem
Shrinking
Projection Method
Hilbert space
Projection
Weak and Strong Convergence
Line Search
Monotonicity
Numerical Experiment
Sufficient
Decrease
Subset
Experiments
Class
Projection method
Equilibrium problem
Strategy
Numerical experiment

Keywords

  • Extragradient methods
  • Non-monotone equilibrium problems
  • Shrinking projection methods
  • Strong convergence
  • Weak convergence

Cite this

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title = "A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces",
abstract = "A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported.",
keywords = "Extragradient methods, Non-monotone equilibrium problems, Shrinking projection methods, Strong convergence, Weak convergence",
author = "Strodiot, {Jean Jacques} and Vuong, {Phan Tu} and Nguyen, {Thi Thu Van}",
year = "2016",
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AU - Vuong, Phan Tu

AU - Nguyen, Thi Thu Van

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N2 - A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported.

AB - A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported.

KW - Extragradient methods

KW - Non-monotone equilibrium problems

KW - Shrinking projection methods

KW - Strong convergence

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