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Averaged Mappings and the Gradient-Projection Algorithm

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  • Hong-Kun Xu

    (National Sun Yat-sen University
    King Saud University)

Abstract

It is well known that the gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this article, we first provide an alternative averaged mapping approach to the GPA. This approach is operator-oriented in nature. Since, in general, in infinite-dimensional Hilbert spaces, GPA has only weak convergence, we provide two modifications of GPA so that strong convergence is guaranteed. Regularization is also applied to find the minimum-norm solution of the minimization problem under investigation.

Suggested Citation

  • Hong-Kun Xu, 2011. "Averaged Mappings and the Gradient-Projection Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 360-378, August.
    • Handle: RePEc:spr:joptap:v:150:y:2011:i:2:d:10.1007_s10957-011-9837-z
    DOI: 10.1007/s10957-011-9837-z
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    References listed on IDEAS

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    1. Han, Deren & Lo, Hong K., 2004. "Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities," European Journal of Operational Research, Elsevier, vol. 159(3), pages 529-544, December.
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    Cited by:

    1. Taksaporn Sirirut & Pattanapong Tianchai, 2018. "On Solving of Constrained Convex Minimize Problem Using Gradient Projection Method," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2018, pages 1-10, October.
    2. Haiyun Zhou & Peiyuan Wang, 2014. "A Simpler Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 716-727, June.
    3. Peichao Duan & Xubang Zheng & Jing Zhao, 2018. "Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems," Mathematics, MDPI, vol. 7(1), pages 1-14, December.
    4. Yuanheng Wang & Tiantian Xu & Jen-Chih Yao & Bingnan Jiang, 2022. "Self-Adaptive Method and Inertial Modification for Solving the Split Feasibility Problem and Fixed-Point Problem of Quasi-Nonexpansive Mapping," Mathematics, MDPI, vol. 10(9), pages 1-15, May.
    5. Lu-Chuan Ceng & Sy-Ming Guu & Jen-Chih Yao, 2014. "Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense," Journal of Global Optimization, Springer, vol. 60(4), pages 617-634, December.
    6. Suthep Suantai & Kunrada Kankam & Prasit Cholamjiak, 2021. "A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting," Mathematics, MDPI, vol. 9(8), pages 1-14, April.
    7. Che, Haitao & Li, Meixia, 2016. "The conjugate gradient method for split variational inclusion and constrained convex minimization problems," Applied Mathematics and Computation, Elsevier, vol. 290(C), pages 426-438.
    8. Ming Tian & Min-Min Li, 2014. "A General Iterative Method for Solving Constrained Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 202-207, July.
    9. Rattanakorn Wattanataweekul & Kobkoon Janngam & Suthep Suantai, 2023. "A Novel Two-Step Inertial Viscosity Algorithm for Bilevel Optimization Problems Applied to Image Recovery," Mathematics, MDPI, vol. 11(16), pages 1-20, August.
    10. Songnian He & Qiao-Li Dong, 2018. "The Combination Projection Method for Solving Convex Feasibility Problems," Mathematics, MDPI, vol. 6(11), pages 1-13, November.
    11. Ali Abkar & Elahe Shahrosvand & Azizollah Azizi, 2017. "The Split Common Fixed Point Problem for a Family of Multivalued Quasinonexpansive Mappings and Totally Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces," Mathematics, MDPI, vol. 5(1), pages 1-18, February.
    12. Yuanheng Wang & Mingyue Yuan & Bingnan Jiang, 2021. "Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems," Mathematics, MDPI, vol. 9(13), pages 1-13, July.
    13. Seifu Endris Yimer & Poom Kumam & Anteneh Getachew Gebrie & Rabian Wangkeeree, 2019. "Inertial Method for Bilevel Variational Inequality Problems with Fixed Point and Minimizer Point Constraints," Mathematics, MDPI, vol. 7(9), pages 1-21, September.
    14. Bingnan Jiang & Yuanheng Wang & Jen-Chih Yao, 2021. "Multi-Step Inertial Regularized Methods for Hierarchical Variational Inequality Problems Involving Generalized Lipschitzian Mappings," Mathematics, MDPI, vol. 9(17), pages 1-20, August.
    15. Ming Tian & Meng-Ying Tong, 2019. "Extension and Application of the Yamada Iteration Algorithm in Hilbert Spaces," Mathematics, MDPI, vol. 7(3), pages 1-13, February.
    16. Q. L. Dong & J. Z. Huang & X. H. Li & Y. J. Cho & Th. M. Rassias, 2019. "MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications," Journal of Global Optimization, Springer, vol. 73(4), pages 801-824, April.
    17. Bing Tan & Shanshan Xu & Songxiao Li, 2020. "Modified Inertial Hybrid and Shrinking Projection Algorithms for Solving Fixed Point Problems," Mathematics, MDPI, vol. 8(2), pages 1-12, February.
    18. Suthep Suantai & Kunrada Kankam & Prasit Cholamjiak, 2020. "A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces," Mathematics, MDPI, vol. 8(1), pages 1-13, January.
    19. Suthep Suantai & Narin Petrot & Montira Suwannaprapa, 2019. "Iterative Methods for Finding Solutions of a Class of Split Feasibility Problems over Fixed Point Sets in Hilbert Spaces," Mathematics, MDPI, vol. 7(11), pages 1-21, October.
    20. Dang Hieu & Pham Kim Quy, 2023. "One-Step iterative method for bilevel equilibrium problem in Hilbert space," Journal of Global Optimization, Springer, vol. 85(2), pages 487-510, February.

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