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A New Projection-type Method with Nondecreasing Adaptive Step-sizes for Pseudo-monotone Variational Inequalities

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Abstract

We propose a new projection-type method with inertial extrapolation for solving pseudo-monotone and Lipschitz continuous variational inequalities in Hilbert spaces. The proposed method does not require the knowledge of the Lipschitz constant as well as the sequential weak continuity of the corresponding operator. We introduce a self-adaptive procedure, which generates dynamic step-sizes converging to a positive constant. It is proved that the sequence generated by the proposed method converges weakly to a solution of the considered variational inequality with the nonasymptotic O(1/n) convergence rate. Moreover, the linear convergence is established under strong pseudo-monotonicity and Lipschitz continuity assumptions. Numerical a exmples for solving a class of Nash–Cournot oligopolistic market equilibrium model and a network equilibrium flow problem are given illustrating the efficiency of the proposed method.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors are grateful to the Editors and the anonymous referees for their constructive comments, which helped improve the presentation of this paper.

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Authors and Affiliations

  1. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Duong Viet Thong

  2. Mathematical Sciences School, University of Southampton, SO17 1BJ, Southampton, UK

    Phan Tu Vuong

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

    Pham Ky Anh

  4. TIMAS, Thang Long University, Hanoi, Vietnam

    Le Dung Muu

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  1. Duong Viet Thong
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  2. Phan Tu Vuong
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  4. Le Dung Muu
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Correspondence to Duong Viet Thong.

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Thong, D.V., Vuong, P.T., Anh, P.K. et al. A New Projection-type Method with Nondecreasing Adaptive Step-sizes for Pseudo-monotone Variational Inequalities. Netw Spat Econ 22, 803–829 (2022). https://doi.org/10.1007/s11067-022-09568-7

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