Split variational inequality problems (SVIPs) have become a comprehensive framework for representing intricate coupled systems in fields such as optimization, machine learning, signal processing, and inverse problems. In this study, we present three innovative Mann-type iterative methods that integrate self-adjusting step-size selection and inertial dynamics to address SVIPs. Theoretically, we proved weak convergence theorems for each of these algorithms under mild and standard conditions, expanding the traditional results by permitting inertial parameters to surpass the usual range of [0, 1) without necessitating restrictive on-line rules. The algorithms were developed to overcome significant computational hurdles in large-scale nonlinear models, and were particularly effective for data-driven applications. Inspired by practical biomedical classification challenges, we incorporated these algorithms into the training process of the Extreme Learning Machine (ELM) to determine optimal output weights with enhanced numerical stability and precision. This improvement significantly boosted the predictive capabilities of ELM models for early disease detection. Empirical tests on four standard medical datasets, heart disease, lung cancer, heart failure, and prostate cancer, showed notable improvements in classification accuracy and robustness. The blend of relaxed inertial conditions, adaptive step-size updating, and Mann-type structures provided a versatile and efficient framework for solving SVIPs and advancing machine learning-based disease diagnostics.
Citation: Lovelyn Madu, Olaniyi Iyiola, Bruce Wade. Fast and efficient optimization algorithms for split variational inequality problem with applications in data classification problems[J]. Journal of Industrial and Management Optimization, 2026, 22(8): 3605-3667. doi: 10.3934/jimo.2026131
Split variational inequality problems (SVIPs) have become a comprehensive framework for representing intricate coupled systems in fields such as optimization, machine learning, signal processing, and inverse problems. In this study, we present three innovative Mann-type iterative methods that integrate self-adjusting step-size selection and inertial dynamics to address SVIPs. Theoretically, we proved weak convergence theorems for each of these algorithms under mild and standard conditions, expanding the traditional results by permitting inertial parameters to surpass the usual range of [0, 1) without necessitating restrictive on-line rules. The algorithms were developed to overcome significant computational hurdles in large-scale nonlinear models, and were particularly effective for data-driven applications. Inspired by practical biomedical classification challenges, we incorporated these algorithms into the training process of the Extreme Learning Machine (ELM) to determine optimal output weights with enhanced numerical stability and precision. This improvement significantly boosted the predictive capabilities of ELM models for early disease detection. Empirical tests on four standard medical datasets, heart disease, lung cancer, heart failure, and prostate cancer, showed notable improvements in classification accuracy and robustness. The blend of relaxed inertial conditions, adaptive step-size updating, and Mann-type structures provided a versatile and efficient framework for solving SVIPs and advancing machine learning-based disease diagnostics.
| [1] |
A. Gibali, Y. Shehu, An efficient iterative method for finding common fixed points and variational inequalities in Hilbert spaces, Optim., 68 (2019), 13–32. https://doi.org/10.1080/02331934.2018.1490417 doi: 10.1080/02331934.2018.1490417
|
| [2] | G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody, 12 (1976), 747–756. |
| [3] |
J. Fan, L. Liu, X. Qin, A subgradient extragradient algorithm with inertial effects for solving strongly pseudomonotone variational inequalities, Optim., 69 (2020), 2199–2215. https://doi.org/10.1080/02331934.2019.1625355 doi: 10.1080/02331934.2019.1625355
|
| [4] |
D. V. Thong, D. V. Hieu, Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J. Comput. Appl. Math., 341 (2018), 80–98. https://doi.org/10.1016/j.cam.2018.03.019 doi: 10.1016/j.cam.2018.03.019
|
| [5] |
Y. Shehu, O. S. Iyiola, S. Reich, A modified inertial subgradient extragradient method for solving variational inequalities, Optim. Eng., 23 (2021), 421–449. https://doi.org/10.1007/s11081-020-09593-w doi: 10.1007/s11081-020-09593-w
|
| [6] |
X. Chang, S. Liu, Z. Deng, S. Li, An inertial subgradient extragradient algorithm with adaptive stepsizes for variational inequality problems, Optim. Methods Softw., 37 (2021), 1507–1526. https://doi.org/10.1080/10556788.2021.1910946 doi: 10.1080/10556788.2021.1910946
|
| [7] |
L. C. Ceng, Q. H. Ansari, J. C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116–2125. https://doi.org/10.1016/j.na.2011.10.012 doi: 10.1016/j.na.2011.10.012
|
| [8] |
L. C. Ceng, A. Petrusel, X. Qin, J. C. Yao, Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints, Optim., 70 (2021), 1337–1358. https://doi.org/10.1080/02331934.2020.1858832 doi: 10.1080/02331934.2020.1858832
|
| [9] |
Y. Yao, O. S. Iyiola, Y. Shehu, Subgradient extragradient method with double inertial steps for variational inequalities, J. Sci. Comput., 90 (2022), 71. https://doi.org/10.1007/s10915-021-01751-1 doi: 10.1007/s10915-021-01751-1
|
| [10] |
I. O. Ibiam, L. O. Madu, E. U. Ofoedu, C. E. Onyi, H. Zegeye, Hybrid Algorithm for Nonlinear Equilibrium, Variational Inequality, and Fixed Point Problems, Basic Phys. Res., 7 (2017), 1–17. https://doi.org/10.5281/zenodo.4249212 doi: 10.5281/zenodo.4249212
|
| [11] |
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in product space, Numer. Algor., 8 (1994), 117722. https://doi.org/10.1007/BF02142692 doi: 10.1007/BF02142692
|
| [12] |
A. R. Eze, M. U. Wickramasinghe, T O. Alakoya, O. T. Mewomo, O. S. Iyiola, An enhanced Mann-type algorithm for split equality monotone variational inclusions and fixed point problems: Applications in image recovery, J. Comput. Appl. Math., 487 (2026), 0377–0427. https://doi.org/10.1016/j.cam.2026.117722 doi: 10.1016/j.cam.2026.117722
|
| [13] |
C. Byrne, A unified treatment for some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120. https://doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
|
| [14] |
Y. Censor, A. Gibali, S. Reich, Algorithms for the split variational inequality problem, Numer. Algor., 59 (2012), 301–323. https://doi.org/10.1007/s11075-011-9490-5 doi: 10.1007/s11075-011-9490-5
|
| [15] | O. S. Iyiola, M. U. Wickramasinghe, T. O. Alakoya, O. T. Mewomo, O. Ogunleye, A New Popov's Method for Solving Quasimonotone Variational Inequalities with Applications in Optimal Control Problems and Machine Learning Algorithms, Netw. Spat. Econ., 2026. https://doi.org/10.1007/s11067-026-09738-x |
| [16] |
C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), 441–453. https://doi.org/10.1088/0266-5611/18/2/310 doi: 10.1088/0266-5611/18/2/310
|
| [17] |
J. M. Hendrickx, A. Olshevsky, Matrix p-norms are NP-hard to approximate if $p \neq 1, 2, \infty$, SIAM J. Matrix Anal. Appl., 31 (2010), 2802–2812. https://doi.org/10.1137/09076773X doi: 10.1137/09076773X
|
| [18] |
G. López, V. Martín-Márquez, F. Wang, H.-K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., 28 (2012), 085004. https://doi.org/10.1088/0266-5611/28/8/085004 doi: 10.1088/0266-5611/28/8/085004
|
| [19] |
J.-Z. Chen, L.-C. Ceng, Y.-Q. Qiu, Z.-R. Kong, Extragradient methods for solving split feasibility and fixed point problems, Fixed Point Theory Appl., 2015 (2015), 192. https://doi.org/10.1186/s13663-015-0441-z doi: 10.1186/s13663-015-0441-z
|
| [20] |
Q. L. Dong, Y. C. Tang, Y. J. Cho, Th. M. Rassias, "Optimal" choice of the step length of the projection and contraction methods for solving the split feasibility problem, J. Glob. Optim., 71 (2018), 341–360 https://doi.org/10.1007/s10898-018-0628-z doi: 10.1007/s10898-018-0628-z
|
| [21] |
A. Gibali, L.-W. Liu, Y.-C. Tang, Note on the modified relaxation CQ algorithm for the split feasibility problem, Optim. Lett., 12 (2018), 817–830. https://doi.org/10.1007/s11590-017-1148-3 doi: 10.1007/s11590-017-1148-3
|
| [22] |
R. Kraikaew, S. Saejung, A simple look at the method for solving split feasibility problems in Hilbert spaces, RACSAM, 114 (2020), 117. https://doi.org/10.1007/s13398-020-00851-1 doi: 10.1007/s13398-020-00851-1
|
| [23] |
B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Probl., 21 (2005), 1655–1665. https://doi.org/10.1088/0266-5611/21/5/009 doi: 10.1088/0266-5611/21/5/009
|
| [24] |
X. Qin, L. Wang, A fixed point method for solving a split feasibility problem in Hilbert spaces, RACSAM, 113 (2019), 315–325. https://doi.org/10.1007/s13398-017-0476-6 doi: 10.1007/s13398-017-0476-6
|
| [25] |
Y. Tang, A. Gibali, Several inertial methods for solving split convex feasibilities and related problems, RACSAM, 114 (2020), 121. https://doi.org/10.1007/s13398-020-00857-9 doi: 10.1007/s13398-020-00857-9
|
| [26] |
H. K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl., 26 (2010), 105018. https://doi.org/10.1088/0266-5611/26/10/105018 doi: 10.1088/0266-5611/26/10/105018
|
| [27] |
Y. Dang, J. Sun, H. Xu, Inertial accelerated algorithms for solving a split feasibility problem, J. Ind. Manag. Optim., 13 (2017), 1383–1394. https://doi.org/10.3934/jimo.2016078 doi: 10.3934/jimo.2016078
|
| [28] |
W. Cholamjiak, Y. Shehu, J. Yao, Prediction of breast cancer through fast optimization techniques applied to machine learning, Optim., 74 (2025), 3629–3657. https://doi.org/10.1080/02331934.2024.2385646 doi: 10.1080/02331934.2024.2385646
|
| [29] |
Q. L. Dong, H. B. Yuan, Y. J. Cho, Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87–102. https://doi.org/10.1007/s11590-016-1102-9 doi: 10.1007/s11590-016-1102-9
|
| [30] |
D. A. Lorenz, T. Pock, An inertial forward–backward algorithm for monotone inclusions, J. Math. Imaging Vis., 51 (2015), 311–325. https://doi.org/10.1007/s10851-014-0523-2 doi: 10.1007/s10851-014-0523-2
|
| [31] |
A. Moudafi, M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 155 (2003), 447–454. https://doi.org/10.1016/S0377-0427(02)00906-8 doi: 10.1016/S0377-0427(02)00906-8
|
| [32] |
P. E. Maingé, Inertial iterative process for fixed points of certain quasi-nonexpansive mappings, Set-Valued Anal., 15 (2007), 67–79. https://doi.org/10.1007/s11228-006-0027-3 doi: 10.1007/s11228-006-0027-3
|
| [33] | Y. Nesterov, A method for solving the convex programming problem with convergence rate $O(1/k^2)$, Dokl. Akad. Nauk SSSR, 269 (1983), 543–547. |
| [34] |
B. T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R Comput. Math. Phys., 4 (1964), 1–17. https://doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
|
| [35] |
T. O. Alakoya, A. Eze, O. S. Iyiola, O. Ugwu, An efficient single projection scheme for solving bilevel variational inequality problems: applications to fractional programming, Results Math., 80 (2025), 214. https://doi.org/10.1007/s00025-025-02528-w doi: 10.1007/s00025-025-02528-w
|
| [36] |
O. T. Mewomo, T. O. Alakoya, A. Eze, O. S. Iyiola, An inertial-like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces, Math. Meth. Appl. Sci., 47 (2024), 9637–9668. https://doi.org/10.1002/mma.10087 doi: 10.1002/mma.10087
|
| [37] |
O. Ogunleye, O. T. Mewomo, T. O. Alakoya, O. S. Iyiola, On fixed point algorithms for solving split inverse problems with applications, Appl. Set-Valued Anal. Optim., 5 (2023), 451–476. https://doi.org/10.23952/asvao.5.2023.3.08 doi: 10.23952/asvao.5.2023.3.08
|
| [38] |
M. U. Wickramasinghe, O. T. Mewomo, T. O. Alakoya, O. S. Iyiola, Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces, Appl. Anal., 103 (2024), 1118–1148. https://doi.org/10.1080/00036811.2023.2233977 doi: 10.1080/00036811.2023.2233977
|
| [39] |
H.-Y. Xu, H.-Y. Lan, Y. Zhao, Y. Ye, Novel extended Halpern-type convergence algorithms for the split common fixed point problem involving $\alpha$-demicontractive operators, Optim., 74 (2025), 343–364. http://doi.org/10.1080/02331934.2023.2256968 doi: 10.1080/02331934.2023.2256968
|
| [40] |
H. He, C. Ling, H. K. Xu, A relaxed projection method for split variational inequalities, J. Optim. Theory Appl., 166 (2015), 213–233. https://doi.org/10.1007/s10957-014-0598-3 doi: 10.1007/s10957-014-0598-3
|
| [41] |
B. Tan, X. Qin, J. C. Yao, Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applications, J. Sci. Comput., 87 (2021), 20. https://doi.org/10.1007/s10915-021-01428-9 doi: 10.1007/s10915-021-01428-9
|
| [42] |
Z. Zhou, B. Tan, S. Li, Inertial algorithms with adaptive stepsizes for split variational inclusion problems and their applications to signal recovery problem, Math. Meth. Appl. Sci., 47 (2024), 9431–9449. https://doi.org/10.1002/mma.9436 doi: 10.1002/mma.9436
|
| [43] |
A. E. Ofem, A. A. Mebawondu, G. C. Ugwunnadi, P. Cholamjiak, O. K. Narain, A novel method for solving split variational inequality and fixed point problems, Appl. Anal., 104 (2025), 3717–3747. https://doi.org/10.1080/00036811.2025.2505615 doi: 10.1080/00036811.2025.2505615
|
| [44] | W. A. Kirk, Fixed point theorems for nonexpansive mappings, Proc. Am. Math. Soc., 16 (1965), 763–766. |
| [45] | H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2 Eds. Springer, New York, 2011. |
| [46] | Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, SIAM, Philadelphia, 1980. Available from: https://epubs.siam.org/doi/book/10.1137/1.9780898719451. |
| [47] | K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990. |
| [48] | W. Rudin, Functional Analysis, 2 Eds., McGraw-Hill, New York, 1991. |
| [49] | Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc., 73 (1967), 591–597. |
| [50] |
J. Wang, Y. Hu, C. Li, J.-C. Yao, Linear convergence of CQ algorithms and applications in gene regulatory network inference, Inverse Probl., 33 (2017), 055017. https://doi.org/10.1088/1361-6420/aa6699 doi: 10.1088/1361-6420/aa6699
|
| [51] |
A. Gibali, D. Mai, N. T. The Vinh, A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications, J. Ind. Manag. Optim., 15 (2019), 963–984. https://doi.org/10.3934/jimo.2018080 doi: 10.3934/jimo.2018080
|
| [52] | S. S. Tai, N. Pholasa, P. Cholamjiak, The modified inertial relaxed CQ algorithm for solving split feasibility problems, J. Ind. Manag. Optim., 14 (2018), 1595–1615. https://doi.org/10.3934/jimo.2018023 |
| [53] |
S. S. Tai, N. Pholasa, P. Cholamjiak, Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems, RACSAM, 113 (2019), 1081-–1099. https://doi.org/10.1007/s13398-018-0535-7 doi: 10.1007/s13398-018-0535-7
|
| [54] |
F. Alvarez, H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3–11. https://doi.org/10.1023/A:1011253113155 doi: 10.1023/A:1011253113155
|
| [55] |
G. N. Ogwo, C. Izuchukwu, O. T. Mewomo, Relaxed inertial methods for solving split variational inequality problems without product space formulation, Acta Math. Sci., 42 (2022), 1701–1733. https://doi.org/10.1007/s10473-022-0501-5 doi: 10.1007/s10473-022-0501-5
|
| [56] |
W. Sun, G. Lu, Y. Jin, Z. Peng, Strong convergence theorems for split variational inequality problems in Hilbert spaces, AIMS Math., 8 (2023), 27291–27308. https://doi.org/10.3934/math.20231396 doi: 10.3934/math.20231396
|
| [57] |
D. R. Han, H. J. He, H. Yang, X. M. Yuan, A customized Douglas–Rachford splitting algorithm for separable convex minimization with linear constraints, Numer. Math., 127 (2014), 167–200. https://doi.org/10.1007/s00211-013-0580-2 doi: 10.1007/s00211-013-0580-2
|
| [58] |
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
|
| [59] | T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2 Eds., Springer, New York, 2009. |
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Lovelyn Madu, Olaniyi Iyiola, Bruce Wade. Fast and efficient optimization algorithms for split variational inequality problem with applications in data classification problems[J]. Journal of Industrial and Management Optimization, 2026, 22(8): 3605-3667. doi: 10.3934/jimo.2026131
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