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Halpern subgradient extragradient method extended to equilibrium problems

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous operators. This paper extends the Halpern subgradient extragradient method (HSEM) to equilibrium problems for pseudomonotone and Lipschitz-type continuous bifunctions. We have replaced naturally two projections in HSEM by two optimization programs and proved the strong convergence of the obtained algorithm. We also present a modification of the proposed algorithm for finding a common solution of an equilibrium problem and a fixed point problem. In several of numerical experiments, we see that the proposed algorithm seems to have a competitive advantage over the extragradient method for equilibrium problems.

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Notes

  1. We chose two diagonal matrices \(Q_1,Q_2\) with their diagonal elements generated randomly in [1, m] and \([-m,0]\), respectively. Next, we made the symmetric positive semidefinite matrix Q by using \(Q_1\) and a random orthogonal matrix. Finally, we made a negative semidefinite T from \(Q_2\) and another random orthogonal matrix, and set \(P=Q-T.\)

  2. \(x\in Fix(T)\) if and only if \(||x-T(x)||=0\).

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Acknowledgments

The author would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper.

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  1. Department of Mathematics, Vietnam National University, Hanoi, 334-Nguyen Trai Street, Hanoi, Vietnam

    Dang Van Hieu

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Van Hieu, D. Halpern subgradient extragradient method extended to equilibrium problems. RACSAM 111, 823–840 (2017). https://doi.org/10.1007/s13398-016-0328-9

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