Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence

Y. Shehu, O.S. Iyiola, Applied Numerical Mathematics 157 (2020) 315–337.

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Journal Article | Epub ahead of print | English

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Author
Shehu, YekiniIST Austria ; Iyiola, Olaniyi S.
Department
Abstract
The projection methods with vanilla inertial extrapolation step for variational inequalities have been of interest to many authors recently due to the improved convergence speed contributed by the presence of inertial extrapolation step. However, it is discovered that these projection methods with inertial steps lose the Fejér monotonicity of the iterates with respect to the solution, which is being enjoyed by their corresponding non-inertial projection methods for variational inequalities. This lack of Fejér monotonicity makes projection methods with vanilla inertial extrapolation step for variational inequalities not to converge faster than their corresponding non-inertial projection methods at times. Also, it has recently been proved that the projection methods with vanilla inertial extrapolation step may provide convergence rates that are worse than the classical projected gradient methods for strongly convex functions. In this paper, we introduce projection methods with alternated inertial extrapolation step for solving variational inequalities. We show that the sequence of iterates generated by our methods converges weakly to a solution of the variational inequality under some appropriate conditions. The Fejér monotonicity of even subsequence is recovered in these methods and linear rate of convergence is obtained. The numerical implementations of our methods compared with some other inertial projection methods show that our method is more efficient and outperforms some of these inertial projection methods.
Publishing Year
Date Published
2020-06-27
Journal Title
Applied Numerical Mathematics
Volume
157
Page
315-337
ISSN
IST-REx-ID

Cite this

Shehu Y, Iyiola OS. Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics. 2020;157:315-337. doi:10.1016/j.apnum.2020.06.009
Shehu, Y., & Iyiola, O. S. (2020). Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics, 157, 315–337. https://doi.org/10.1016/j.apnum.2020.06.009
Shehu, Yekini, and Olaniyi S. Iyiola. “Projection Methods with Alternating Inertial Steps for Variational Inequalities: Weak and Linear Convergence.” Applied Numerical Mathematics 157 (2020): 315–37. https://doi.org/10.1016/j.apnum.2020.06.009.
Y. Shehu and O. S. Iyiola, “Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence,” Applied Numerical Mathematics, vol. 157, pp. 315–337, 2020.
Shehu Y, Iyiola OS. 2020. Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence. Applied Numerical Mathematics. 157, 315–337.
Shehu, Yekini, and Olaniyi S. Iyiola. “Projection Methods with Alternating Inertial Steps for Variational Inequalities: Weak and Linear Convergence.” Applied Numerical Mathematics, vol. 157, Elsevier, 2020, pp. 315–37, doi:10.1016/j.apnum.2020.06.009.
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