Search Advanced

AIMS Mathematics

2024, Volume 9, Issue 4: 8476-8496. doi: 10.3934/math.2024412
Previous Article Next Article
Research article

A new two-step inertial algorithm for solving convex bilevel optimization problems with application in data classification problems

  • 1.
    PhD Degree Program in Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, under the CMU Presidential Scholarship, Thailand
  • 2.
    Research Center in Optimization and Computational Intelligence for Big Data Prediction, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
  • Received: 12 January 2024 Revised: 21 February 2024 Accepted: 23 February 2024 Published: 28 February 2024

  • MSC : 47H10, 65K10, 90C25

  • In this paper, we propose a new accelerated algorithm for solving convex bilevel optimization problems using some fixed point and two-step inertial techniques. Our focus is on analyzing the convergence behavior of the proposed algorithm. We establish a strong convergence theorem for our algorithm under some control conditions. To demonstrate the effectiveness of our algorithm, we utilize it as a machine learning algorithm to solve data classification problems of some noncommunicable diseases, and compare its efficacy with BiG-SAM and iBiG-SAM.

    Citation: Puntita Sae-jia, Suthep Suantai. A new two-step inertial algorithm for solving convex bilevel optimization problems with application in data classification problems[J]. AIMS Mathematics, 2024, 9(4): 8476-8496. doi: 10.3934/math.2024412

    Related Papers:

  • In this paper, we propose a new accelerated algorithm for solving convex bilevel optimization problems using some fixed point and two-step inertial techniques. Our focus is on analyzing the convergence behavior of the proposed algorithm. We establish a strong convergence theorem for our algorithm under some control conditions. To demonstrate the effectiveness of our algorithm, we utilize it as a machine learning algorithm to solve data classification problems of some noncommunicable diseases, and compare its efficacy with BiG-SAM and iBiG-SAM.


    加载中


    [1] P. Thongpaen, W. Inthakon, T. Leerapun, S. Suantai, A new accelerated algorithm for convex Bilevel optimization problems and applications in data classification, Symmetry, 14 (2022), 2617. https://doi.org/10.3390/sym14122617 doi: 10.3390/sym14122617
    [2] K. Janngam, S. Suantai, Y. J. Cho, A. Kaewkhao, A novel inertial viscosity algorithm for Bilevel optimization problems applied to classification problems, Mathematics, 11 (2023), 2617. https://doi.org/10.3390/math11143241 doi: 10.3390/math11143241
    [3] P. Thongsri, B. Panyanak, S. Suantai, A new accelerated algorithm based on fixed point method for convex Bilevel optimization problems with applications, Mathematics, 11 (2023), 702. https://doi.org/10.3390/math11030702 doi: 10.3390/math11030702
    [4] P. Sae-jia, S. Suantai, A novel algorithm for convex Bi-level optimization problems in Hilbert spaces with applications, Thai. J. Math., 21 (2023), 625–645. https://doi.org/10.3390/math11143241 doi: 10.3390/math11143241
    [5] N. Parikh, S. Boyd, Proximal algorithms, Found. Trend. Optim., 1 (2014), 127–239. http://dx.doi.org/10.1561/2400000003 doi: 10.1561/240000000
    [6] W. R. Mann, Mean value methods in iteration, P. Am. Math. Soc., 4 (1953), 506–510.
    [7] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 174–276.
    [8] B. Halpern, Fixed points of nonexpanding maps, Found. Trend. Optim., 73 (1967), 957–961. http://dx.doi.org/10.1561/2400000003 doi: 10.1561/2400000003
    [9] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287–292.
    [10] S. Ishikawa, Fixed points by a new iteration method, P. Am. Math. Soc., 44 (1974), 147–150. https://doi.org/10.2307/2039245 doi: 10.2307/2039245
    [11] A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., 241 (2000), 46–55. https://doi.org/10.1006/jmaa.1999.6615 doi: 10.1006/jmaa.1999.6615
    [12] R. P. Agarwal, D. O. Regan, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex A., 8 (2007), 61.
    [13] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal., 67 (2007), 2350–2360. https://doi.org/10.1155/2010/407651 doi: 10.1155/2010/407651
    [14] W. Takahashi, Viscosity approximation methods for countable family of nonexpansive mapping in Banach spaces, Nonlinear Anal., 70 (2009), 719–734. https://doi.org/10.1016/j.na.2008.01.005 doi: 10.1016/j.na.2008.01.005
    [15] C. Klin-eam, S. Suantai, Strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal., 73 (2010), 431–439. https://doi.org/10.1016/j.na.2010.03.034 doi: 10.1016/j.na.2010.03.034
    [16] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4 (1964), 1–17. https://doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
    [17] A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183–202. https://doi.org/10.1137/080716542 doi: 10.1137/080716542
    [18] J. Puangpee, S. Suantai, A new accelerated viscosity iterative method for an infinite family of nonexpansive mappings with applications to image restoration problems, Mathematics, 8 (2020), 615. https://doi.org/10.3390/math8040615 doi: 10.3390/math8040615
    [19] B. T. Polyak, Introduction to optimization. optimization software, New York: Publications Division, 1987.
    [20] Q. L. Dong, Y. J. Cho, T. M. Rassias, General inertial Mann algorithms and their convergence analysis for nonexpansive mappings, Appl. Nonlinear Anal., 2018,175–191. https://doi.org/10.1007/978-3-319-89815-5_7 doi: 10.1007/978-3-319-89815-5_7
    [21] S. Sabach, S. Shtern, A first order method for solving convex bilevel optimization problems, SIAM J. Optim., 27 (2017), 640–660. https://doi.org/10.1137/16M105592 doi: 10.1137/16M105592
    [22] Y. Shehu, P. T. Vuong, A. Zemkoho, An inertial extrapolation method for convex simple bilevel optimization, Optim. Method. Softw., 36 (2021), 1–19. https://doi.org/10.48550/arXiv.1809.06250 doi: 10.48550/arXiv.1809.06250
    [23] K. Goebel, S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, New York: Marcel Dekker, 1984.
    [24] K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces, J. Nonlinear Convex A., 8 (2007), 11.
    [25] W. Takahashi, Introduction to nonlinear and convex analysis, Yokohama Publishers, 2009.
    [26] L. Bussaban, S. Suantai, A. Kaewkhao, A paralle inertial S-iteration forward-backward algorithm for regression and classification problems, Carpathian J. Math., 36 (2020), 35–44.
    [27] W. Takahashi, Nonlinear functional analysis, Yokohama Publishers, 2000.
    [28] K. Goebel, W. A. Kirk, Topic in metric fixed point theory, Cambridge University Press, 1990.
    [29] K. Aoyam, Y. Yasunori, W. Takahashi, M. Toyoda, On a strongly nonexpansive sequence in a Hilbert space, J. Nonlinear Convex A., 8 (2007), 471–490.
    [30] H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, B. Aust. Math. Soc., 65 (2002), 109–113. https://doi.org/10.1017/S0004972700020116 doi: 10.1017/S0004972700020116
    [31] P. E. Maing, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899–912. https://doi.org/10.1007/s11228-008-0102-z doi: 10.1007/s11228-008-0102-z
    [32] G. B. Huang, Q. Y. Zhu, C. K. Siew, Extreme learning machine: Theory and applications, Neurocomputing, 70 (2006), 489–501. https://doi.org/10.1016/j.neucom.2005.12.126 doi: 10.1016/j.neucom.2005.12.126
    [33] Little Max, Parkinsons, UCI Machine Learning Repository, 2008. Available from: https://archive.ics.uci.edu/dataset/174/parkinsons.
    [34] Kaggle, Diabetes dataset, Kaggle, 1990. Available from: https://www.kaggle.com/datasets/mathchi/diabetes-data-set/data.
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(791) PDF downloads(87) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(2)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog