Double inertial steps extragadient-type methods for solving optimal control and image restoration problems

Austine Efut Ofem; Jacob Ashiwere Abuchu; Godwin Chidi Ugwunnadi; Hossam A. Nabwey; Abubakar Adamu; Ojen Kumar Narain; Austine Efut Ofem; Jacob Ashiwere Abuchu; Godwin Chidi Ugwunnadi; Hossam A. Nabwey; Abubakar Adamu; Ojen Kumar Narain

doi:10.3934/math.2024629

AIMS Mathematics

2024, Volume 9, Issue 5: 12870-12905. doi: 10.3934/math.2024629

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Double inertial steps extragadient-type methods for solving optimal control and image restoration problems

1.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
2.
Department of Mathematics, University of Calabar, Calabar, Nigeria
3.
Department of Mathematics, University of Eswatini, Private Bag 4, Kwaluseni, Eswatini
4.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, Pretoria, South Africa
5.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
6.
Department of Basic Engineering, Faculty of Engineering, Menoufia University, Shibin el Kom 32511, Egypt
7.
Operational Research Center in Healthcare, Near East University, TRNC Mersin 10, Nicosia 99138, Turkey
8.
Mathematics Institute, African University of Science and Technology, Abuja 900107, Nigeria

Received: 12 January 2024 Revised: 19 March 2024 Accepted: 26 March 2024 Published: 07 April 2024
MSC : 47H05, 47J20, 47J25, 65K15

In order to approximate the common solution of quasi-nonexpansive fixed point and pseudo-monotone variational inequality problems in real Hilbert spaces, this paper presented three new modified sub-gradient extragradient-type methods. Our algorithms incorporated viscosity terms and double inertial extrapolations to ensure strong convergence and to speed up convergence. No line search methods of the Armijo type were required by our algorithms. Instead, they employed a novel self-adaptive step size technique that produced a non-monotonic sequence of step sizes while also correctly incorporating a number of well-known step sizes. The step size was designed to lessen the algorithms' reliance on the initial step size. Numerical tests were performed, and the results showed that our step size is more effective and that it guarantees that our methods require less execution time. We stated and proved the strong convergence of our algorithms under mild conditions imposed on the control parameters. To show the computational advantage of the suggested methods over some well-known methods in the literature, several numerical experiments were provided. To test the applicability and efficiencies of our methods in solving real-world problems, we utilized the proposed methods to solve optimal control and image restoration problems.
- variational inequality problem,
- fixed point,
- pseudo-monotone operator,
- strong convergence,
- viscosity,
- subgradient extragradient method
Citation: Austine Efut Ofem, Jacob Ashiwere Abuchu, Godwin Chidi Ugwunnadi, Hossam A. Nabwey, Abubakar Adamu, Ojen Kumar Narain. Double inertial steps extragadient-type methods for solving optimal control and image restoration problems[J]. AIMS Mathematics, 2024, 9(5): 12870-12905. doi: 10.3934/math.2024629

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Abstract

In order to approximate the common solution of quasi-nonexpansive fixed point and pseudo-monotone variational inequality problems in real Hilbert spaces, this paper presented three new modified sub-gradient extragradient-type methods. Our algorithms incorporated viscosity terms and double inertial extrapolations to ensure strong convergence and to speed up convergence. No line search methods of the Armijo type were required by our algorithms. Instead, they employed a novel self-adaptive step size technique that produced a non-monotonic sequence of step sizes while also correctly incorporating a number of well-known step sizes. The step size was designed to lessen the algorithms' reliance on the initial step size. Numerical tests were performed, and the results showed that our step size is more effective and that it guarantees that our methods require less execution time. We stated and proved the strong convergence of our algorithms under mild conditions imposed on the control parameters. To show the computational advantage of the suggested methods over some well-known methods in the literature, several numerical experiments were provided. To test the applicability and efficiencies of our methods in solving real-world problems, we utilized the proposed methods to solve optimal control and image restoration problems.

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