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Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications

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Abstract

In this paper, we introduce two self-adaptive algorithms for solving a class of non-Lipschitz equilibrium problems. These algorithms are very simple in the sense that at each step, they require only one projection onto a feasible set. Their convergence can be established under quite mild assumptions. More precisely, the weak (strong) convergence of the first algorithm is proved under the pseudo-paramonotonicity (strong pseudomonotonicity) conditions, respectively. Especially, the convexity in the second argument of the involving bifunction is not required. In the second algorithm, the weak convergence is established under the pseudomonotonicity. Moreover, it is proved that under some additional conditions, the solvability of the equilibrium problem is equivalent to the boundedness of the sequences generated by the proposed algorithms. Some applications to the optimization problems and variational inequality problems as well as to transport equilibrium problems are also considered.

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Acknowledgements

The authors thank two anonymous referees and the editor for their constructive comments which helped to improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Pham Ky Anh

  2. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

    Trinh Ngoc Hai

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  1. Pham Ky Anh
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Correspondence to Trinh Ngoc Hai.

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Anh, P.K., Hai, T.N. Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications. J Glob Optim 73, 637–657 (2019). https://doi.org/10.1007/s10898-018-0722-2

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  • DOI: https://doi.org/10.1007/s10898-018-0722-2

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