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Analysis of two variants of an inertial projection algorithm for finding the minimum-norm solutions of variational inequality and fixed point problems

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Abstract

We study variational inequalities and fixed point problems in real Hilbert spaces. A new algorithm is proposed for finding a common element of the solution set of a pseudo-monotone variational inequality and the fixed point set of a demicontractive mapping. The advantage of our algorithm is that it does not require prior information regarding the Lipschitz constant of the variational inequality operator and that it only computes one projection onto the feasible set per iteration. In addition, we do not need the sequential weak continuity of the variational inequality operator in order to establish our strong convergence theorem. Next, we also obtain an R-linear convergence rate for a related relaxed inertial gradient method under strong pseudo-monotonicity and Lipschitz continuity assumptions on the variational inequality operator. Finally, we present several numerical examples which illustrate the performance and the effectiveness of our algorithm.

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Acknowledgements

The authors are very grateful to an anonymous referee for several pertinent comments and useful suggestions.

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Authors and Affiliations

  1. Vietnam National University, Ho Chi Minh City, Vietnam

    Ha Manh Linh

  2. Department of Mathematics and Physics, University of Information Technology, Ho Chi Minh City, Vietnam

    Ha Manh Linh

  3. Department of Mathematics, The Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

  4. Faculty of Mathematical Economics, National Economics University, Hanoi City, Vietnam

    Duong Viet Thong & Nguyen Phuong Lan

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

    Vu Tien Dung

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  1. Ha Manh Linh
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  2. Simeon Reich
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  3. Duong Viet Thong
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  4. Vu Tien Dung
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  5. Nguyen Phuong Lan
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Correspondence to Simeon Reich.

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Linh, H.M., Reich, S., Thong, D.V. et al. Analysis of two variants of an inertial projection algorithm for finding the minimum-norm solutions of variational inequality and fixed point problems. Numer Algor 89, 1695–1721 (2022). https://doi.org/10.1007/s11075-021-01169-8

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  • DOI: https://doi.org/10.1007/s11075-021-01169-8

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