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

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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.

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Acknowledgments

The authors would like to thank the two anonymous referees and the Associate Editor for their valuable comments that allowed to improve substantially the original version of this paper. This research is funded by the Department of Science and Technology at Ho Chi Minh City, Vietnam. Computing resources and support provided by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST) are gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Computational Science and Technology-HCMC (ICST), Ho Chi Minh City, Vietnam

    Jean Jacques Strodiot, Phan Tu Vuong & Thi Thu Van Nguyen

  2. University of Namur, Namur, Belgium

    Jean Jacques Strodiot

  3. University of Technical Education of HCMC, Ho Chi Minh City, Vietnam

    Phan Tu Vuong

  4. University of Science, VNU-HCMC, Ho Chi Minh City, Vietnam

    Thi Thu Van Nguyen

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  1. Jean Jacques Strodiot
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  2. Phan Tu Vuong
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  3. Thi Thu Van Nguyen
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Correspondence to Jean Jacques Strodiot.

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Dedicated to Professor Van Hien Nguyen

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Strodiot, J.J., Vuong, P.T. & Nguyen, T.T.V. A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J Glob Optim 64, 159–178 (2016). https://doi.org/10.1007/s10898-015-0365-5

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  • DOI: https://doi.org/10.1007/s10898-015-0365-5

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