Convergence Analysis of Altered Inertial Subgradient Extragradient Method for Solving Equilibrium and Fixed Point Problems in Hilbert Spaces
Main Article Content
Abstract
This paper aims to derive weak convergence analysis of the subgradient extragradient method with an alternating inertial step for addressing equilibrium and fixed-point problems in real Hilbert spaces. Our approach incorporates altered inertial techniques into a Halpern-type method, extending the classical subgradient extragradient framework.
Article Details
References
- G. Long, S. Liu, G. Xu, S.W. Wong, H. Chen, et al., A Perforation-Erosion Model for Hydraulic-Fracturing Applications, SPE Prod. Oper. 33 (2018), 770–783. https://doi.org/10.2118/174959-PA.
- B. Xiao, W. Wang, X. Zhang, G. Long, J. Fan, et al., A Novel Fractal Solution for Permeability and Kozeny-Carman Constant of Fibrous Porous Media Made up of Solid Particles and Porous Fibers, Powder Technol. 349 (2019), 92–98. https://doi.org/10.1016/j.powtec.2019.03.028.
- B. Xiao, X. Zhang, G. Jiang, G. Long, W. Wang, et al., Kozeny–Carman Constant for Gas Flow through Fibrous Porous Media by Fractal-Monte Carlo Simulations, Fractals 27 (2019), 1950062. https://doi.org/10.1142/S0218348X19500622.
- V. Dadashi, O.S. Iyiola, Y. Shehu, The Subgradient Extragradient Method for Pseudomonotone Equilibrium Problems, Optimization 69 (2019), 901–923. https://doi.org/10.1080/02331934.2019.1625899.
- D. Van Hieu, A. Gibali, Strong Convergence of Inertial Algorithms for Solving Equilibrium Problems, Optim. Lett. 14 (2019), 1817–1843. https://doi.org/10.1007/s11590-019-01479-w.
- D. Van Hieu, P.K. Quy, L. Van Vy, Explicit Iterative Algorithms for Solving Equilibrium Problems, Calcolo 56 (2019), 11. https://doi.org/10.1007/s10092-019-0308-5.
- Y. Shehu, O.S. Iyiola, D.V. Thong, N.T.C. Van, An Inertial Subgradient Extragradient Algorithm Extended to Pseudomonotone Equilibrium Problems, Math. Methods Oper. Res. 93 (2020), 213–242. https://doi.org/10.1007/s00186-020-00730-w.
- D.V. Thong, P. Cholamjiak, M.T. Rassias, Y.J. Cho, Strong Convergence of Inertial Subgradient Extragradient Algorithm for Solving Pseudomonotone Equilibrium Problems, Optim. Lett. 16 (2021), 545–573. https://doi.org/10.1007/s11590-021-01734-z.
- Y. Censor, A. Gibali, S. Reich, The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space, J. Optim. Theory Appl. 148 (2010), 318–335. https://doi.org/10.1007/s10957-010-9757-3.
- Y. Censor, A. Gibali, S. Reich, Strong Convergence of Subgradient Extragradient Methods for the Variational Inequality Problem in Hilbert Space, Optim. Methods Softw. 26 (2011), 827–845. https://doi.org/10.1080/10556788.2010.551536.
- K.J. Arrow, G. Debreu, Existence of an Equilibrium for a Competitive Economy, Econometrica 22 (1954), 265. https://doi.org/10.2307/1907353.
- A.A. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richesses, L. Hachette, New York, (1838).
- S. Wernecke, M. D'Addario, Maximum Entropy Image Reconstruction, IEEE Trans. Comput. C-26 (1977), 351–364. https://doi.org/10.1109/TC.1977.1674845.
- D. Kundur, D. Hatzinakos, Blind Image Deconvolution, IEEE Signal Process. Mag. 13 (1996), 43–64. https://doi.org/10.1109/79.489268.
- S. Gharehkhani, E. Sadeghinezhad, S.N. Kazi, H. Yarmand, A. Badarudin, et al., Basic Effects of Pulp Refining on Fiber Properties—A Review, Carbohydr. Polym. 115 (2015), 785–803. https://doi.org/10.1016/j.carbpol.2014.08.047.
- A. Nagurney, Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, 1993.
- M. Patriksson, Traffic Assignment Problem: Models and Methods, Dover Publications, 2015.
- J. Nash, Non-Cooperative Games, Ann. Math. 54 (1951), 286. https://doi.org/10.2307/1969529.
- K. Fan, A Minimax Inequality and Applications, in: O. Shisha, (ed) Inequalities III, Academic Press, New York, (1972).
- H. Nikaid^{o}, K. Isoda, Note on Non-Cooperative Convex Game, Pac. J. Math. 5 (1955), 807–815. https://doi.org/10.2140/pjm.1955.5.807.
- J. Gwinner, On the Penalty Method for Constrained Variational Inequalities, in: J.-B. Hiriart-Urruty, W. Oettli, J. Stoer, (eds) Optimization - Theory and Algorithms, Marcel Dekker, New York, pp. 197–211, (1981).
- P.T. Vuong, J.J. Strodiot, V.H. Nguyen, On Extragradient-Viscosity Methods for Solving Equilibrium and Fixed Point Problems in a Hilbert Space, Optimization 64 (2013), 429–451. https://doi.org/10.1080/02331934.2012.759327.
- D. Van Hieu, Halpern Subgradient Extragradient Method Extended to Equilibrium Problems, Rev. Real Acad. Cienc. Exactas F'{i}s. Nat. Ser. A Mat. RACSAM 111 (2016), 823–840. https://doi.org/10.1007/s13398-016-0328-9.
- S. Saejung, P. Yotkaew, Approximation of Zeros of Inverse Strongly Monotone Operators in Banach Spaces, Nonlinear Anal. Theory Methods Appl. 75 (2012), 742–750. https://doi.org/10.1016/j.na.2011.09.005.
- B. Panyanak, C. Khunpanuk, N. Pholasa, N. Pakkaranang, Dynamical Inertial Extragradient Techniques for Solving Equilibrium and Fixed-Point Problems in Real Hilbert Spaces, J. Inequal. Appl. 2023 (2023), 7. https://doi.org/10.1186/s13660-023-02912-6.
- D. Quoc Tran, M. Le Dung, V.H. Nguyen, Extragradient Algorithms Extended to Equilibrium Problems, Optimization 57 (2008), 749–776. https://doi.org/10.1080/02331930601122876.
- P.T. Harker, J.S. Pang, A Damped-Newton Method for the Linear Complementarity Problems, in: E.L. Allgower, K. Georg, (eds) Computational Solution of Nonlinear Systems of Equations, Lectures in Applied Mathematics, Vol 26, American Mathematical Society, Providence, RI, pp. 265–284, (1990).
- D.V. Hieu, Strong Convergence of a New Hybrid Algorithm for Fixed Point Problems and Equilibrium Problems, Math. Model. Anal. 24 (2018), 1–19. https://doi.org/10.3846/mma.2019.001.
- B. Tan, G.Y. Cho, J.C. Yao, Accelerated Inertial Subgradient Extragradient Algorithms with Non-monotonic Step Sizes for Equilibrium Problems and Fixed Point Problems, J. Nonlinear Var. Anal. 6 (2022), 89–122. https://doi.org/10.23952/jnva.6.2022.1.06.
- X. Qin, N.T. An, Smoothing Algorithms for Computing the Projection onto a Minkowski Sum of Convex Sets, Comput. Optim. Appl. 74 (2019), 821–850. https://doi.org/10.1007/s10589-019-00124-7.
- N.T. An, N.M. Nam, X. Qin, Solving k-Center Problems Involving Sets Based on Optimization Techniques, J. Glob. Optim. 76 (2019), 189–209. https://doi.org/10.1007/s10898-019-00834-6.
- D.R. Sahu, J.C. Yao, M. Verma, K.K. Shukla, Convergence Rate Analysis of Proximal Gradient Methods with Applications to Composite Minimization Problems, Optimization 70 (2020), 75–100. https://doi.org/10.1080/02331934.2019.1702040.
- H. Iiduka, I. Yamada, A Subgradient-Type Method for the Equilibrium Problem over the Fixed Point Set and Its Applications, Optimization 58 (2009), 251–261. https://doi.org/10.1080/02331930701762829.
- P.-E. Maingé, A Hybrid Extragradient-Viscosity Method for Monotone Operators and Fixed Point Problems, SIAM J. Control Optim. 47 (2008), 1499–1515. https://doi.org/10.1137/060675319.
- P.E. Mainge, A. Moudafi, Coupling Viscosity Methods with the Extragradient Algorithm for Solving Equilibrium Problems, J. Nonlinear Convex Anal. 9 (2008), 283–294.
- S.D. Flr{a}m, A.S. Antipin, Equilibrium Programming Using Proximal-like Algorithms, Math. Program. 78 (1996), 29–41. https://doi.org/10.1007/BF02614504.
- D. Quoc Tran, M. Le Dung, V.H. Nguyen, Extragradient Algorithms Extended to Equilibrium Problems, Optimization 57 (2008), 749–776. https://doi.org/10.1080/02331930601122876.
- G.M. Korpelevich, An Extragradient Method for Finding Saddle Points and Other Problems, Ekon Mat. Metody 12 (1976), 747–756. https://cir.nii.ac.jp/crid/1571698600143951616.
- M. Bianchi, S. Schaible, Generalized Monotone Bifunctions and Equilibrium Problems, J. Optim. Theory Appl. 90 (1996), 31–43. https://doi.org/10.1007/bf02192244.
- H. Xu, Viscosity Approximation Methods for Nonexpansive Mappings, J. Math. Anal. Appl. 298 (2004), 279–291. https://doi.org/10.1016/j.jmaa.2004.04.059.