Accelerated non-monotonic explicit proximal-type method for solving equilibrium programming with convex constraints and its applications

Pongsakorn Yotkaew; Nopparat Wairojjana; Nuttapol Pakkaranang; Pongsakorn Yotkaew; Nopparat Wairojjana; Nuttapol Pakkaranang

doi:10.3934/math.2021622

AIMS Mathematics

2021, Volume 6, Issue 10: 10707-10727. doi: 10.3934/math.2021622

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Accelerated non-monotonic explicit proximal-type method for solving equilibrium programming with convex constraints and its applications

1.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2.
Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage, 1 Moo 20 Phaholyothin Rd., Klong Neung, Klong Luang, Pathumthani, 13180, Thailand
3.
Department of Mathematics, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand

Received: 19 January 2021 Accepted: 20 July 2021 Published: 23 July 2021
MSC : 47H05, 47H10, 65K15, 65Y05, 68W10

The main objective of this study is to introduce a new two-step proximal-type method to solve equilibrium problems in a real Hilbert space. This problem is a general mathematical model and includes a number of mathematical problems as a special case, such as optimization problems, variational inequalities, fixed point problems, saddle time problems and Nash equilibrium point problems. A new method is analogous to the famous two-step extragradient method that was used to solve variational inequality problems in a real Hilbert space established previously. The proposed iterative method uses an inertial scheme and a new non-monotone stepsize rule based on local bifunctional values rather than any line search method. A strong convergence theorem for the constructed method is proven by letting mild conditions on a bifunction. These results are being used to solve fixed point problems as well as variational inequalities. Finally, we considered two test problems, and the computational performance was presented to show the performance and efficiency of the proposed method.
- strong convergence,
- Lipschitz-type condition,
- equilibrium problem,
- variational inequalities,
- fixed point problems
Citation: Pongsakorn Yotkaew, Nopparat Wairojjana, Nuttapol Pakkaranang. Accelerated non-monotonic explicit proximal-type method for solving equilibrium programming with convex constraints and its applications[J]. AIMS Mathematics, 2021, 6(10): 10707-10727. doi: 10.3934/math.2021622

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Abstract

The main objective of this study is to introduce a new two-step proximal-type method to solve equilibrium problems in a real Hilbert space. This problem is a general mathematical model and includes a number of mathematical problems as a special case, such as optimization problems, variational inequalities, fixed point problems, saddle time problems and Nash equilibrium point problems. A new method is analogous to the famous two-step extragradient method that was used to solve variational inequality problems in a real Hilbert space established previously. The proposed iterative method uses an inertial scheme and a new non-monotone stepsize rule based on local bifunctional values rather than any line search method. A strong convergence theorem for the constructed method is proven by letting mild conditions on a bifunction. These results are being used to solve fixed point problems as well as variational inequalities. Finally, we considered two test problems, and the computational performance was presented to show the performance and efficiency of the proposed method.

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