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Strongly convergent inertial proximal point method with correction terms: Theory, applications in machine learning for Air Quality prediction

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  • Ogwo, Grace Nnennaya
  • Cholamjiak, Watcharaporn
  • Shehu, Yekini

Abstract

This paper introduces a novel Halpern-type proximal point algorithm that incorporates both inertial and correction terms for solving monotone inclusion problems in real Hilbert spaces. The inertial component accelerates convergence, while the correction term enhances numerical stability and dampens oscillations. Under standard assumptions, we establish strong convergence to a solution. Beyond qualitative convergence, we derive explicit convergence rates: with αk=1/(k+1), we prove ∑i=1k‖yi−xi−1‖2=O(logk) and min1≤i≤k‖yi−wi−1‖2=O(logk/k), providing quantitative performance guarantees. To demonstrate practical utility, we implement the algorithm within an extreme learning machine (ELM) framework to forecast urban benzene (C6H6) concentrations using multisensor air-quality data. The numerical experiments confirm the algorithm’s effectiveness in high-dimensional machine learning applications, highlighting its robustness, stability, and computational efficiency.

Suggested Citation

  • Ogwo, Grace Nnennaya & Cholamjiak, Watcharaporn & Shehu, Yekini, 2026. "Strongly convergent inertial proximal point method with correction terms: Theory, applications in machine learning for Air Quality prediction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 246(C), pages 547-569.
    • Handle: RePEc:eee:matcom:v:246:y:2026:i:c:p:547-569
    DOI: 10.1016/j.matcom.2026.02.016
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    References listed on IDEAS

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    1. O. Boikanyo & G. Moroşanu, 2011. "Inexact Halpern-type proximal point algorithm," Journal of Global Optimization, Springer, vol. 51(1), pages 11-26, September.
    2. Fenghui Wang & Huanhuan Cui, 2012. "On the contraction-proximal point algorithms with multi-parameters," Journal of Global Optimization, Springer, vol. 54(3), pages 485-491, November.
    3. Damek Davis & Wotao Yin, 2017. "Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 783-805, August.
    4. Boţ, Radu Ioan & Csetnek, Ernö Robert & Hendrich, Christopher, 2015. "Inertial Douglas–Rachford splitting for monotone inclusion problems," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 472-487.
    5. Hadi Khatibzadeh & Sajad Ranjbar, 2013. "On the Strong Convergence of Halpern Type Proximal Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 385-396, August.
    6. Hedy Attouch & Alexandre Cabot, 2020. "Convergence rate of a relaxed inertial proximal algorithm for convex minimization," Post-Print hal-02415789, HAL.
    7. Cholamjiak, Watcharaporn & Dutta, Hemen, 2022. "Viscosity modification with parallel inertial two steps forward-backward splitting methods for inclusion problems applied to signal recovery," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    8. Lateef O. Jolaoso & Yekini Shehu & Jen-Chih Yao, 2024. "Strongly Convergent Inertial Proximal Point Algorithm Without On-line Rule," Journal of Optimization Theory and Applications, Springer, vol. 200(2), pages 555-584, February.
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