An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings

Abstract

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.

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Acknowledgements

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The second author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the IMU and NRF.

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Correspondence to Oluwatosin Temitope Mewomo.

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Alakoya, T.O., Taiwo, A., Mewomo, O.T. et al. An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings. Ann Univ Ferrara (2021). https://doi.org/10.1007/s11565-020-00354-2

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Keywords

  • Inertial algorithm
  • Subgradient extragradient method
  • Variational inequality problems
  • Fixed point problems
  • Nonexpansive mappings
  • Equilibrium problems
  • Convex minimization problems
  • Zeros problems

Mathematics Subject Classification

  • 65K15
  • 47J25
  • 65J15
  • 90C33