Advancing optimization algorithms with fixed point theory in generalized metric vector spaces
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Abstract
This research develops and evaluates an adaptive parameter-based fixed point iterative algorithm within generalized metric vector spaces to improve stability and convergence speed in optimization problems. The study extends fixed point theory beyond classical metric spaces by incorporating a more flexible structure that accommodates non-Euclidean systems, commonly found in machine learning, data analysis, and dynamic systems optimization. The proposed adaptive fixed point algorithm modifies the conventional iterative method: where the adaptive parameter dynamically adjusts based on the previous iterations: with as a control constant. A numerical case study demonstrates the algorithm’s effectiveness, comparing it with the classical Banach Fixed Point Theorem. Results show that the adaptive method requires fewer iterations to achieve convergence while maintaining higher stability, significantly outperforming the standard approach. The findings suggest that incorporating adaptive parameters in fixed point iterations enhances computational efficiency, particularly in non-convex optimization and deep learning training models. Future research will explore the algorithm’s robustness in high-dimensional spaces, its integration with hybrid optimization techniques, and applications in uncertain and noisy environments.
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How to Cite
Vinsensia, D., Utami, Y., Awawdeh, B. K., & Bausch, N. B. (2024). Advancing optimization algorithms with fixed point theory in generalized metric vector spaces. International Journal of Basic and Applied Science, 13(3), 146–156. https://doi.org/10.35335/ijobas.v13i3.621
References
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J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables. SIAM, 2000. [Online]. Available: https://epubs.siam.org/doi/pdf/10.1137/1.9780898719468.bm
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H. H. Bauschke, P. L. Combettes, H. H. Bauschke, and P. L. Combettes, “Correction to: convex analysis and monotone operator theory in Hilbert spaces,” in Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2017, pp. 1283–1306. doi: https://doi.org/10.1007/978-3-319-48311-5_31.
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S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundam. Math., vol. 3, no. 1, pp. 133–181, 1922.
K. Goebel and W. A. Kirk, “Fixed point theory in metric spaces,” in Topics in Fixed Point Theory, Cambridge University Press, Cambridge, 1990, pp. 101–158. doi: https://doi.org/10.1007/978-3-319-01586-6_4.
E. K. Ryu and S. Boyd, “Primer on monotone operator methods,” in Appl. comput. math, vol. 15, no. 1, 2016, pp. 3–43. [Online]. Available: https://web.stanford.edu/~boyd/papers/pdf/monotone_primer.pdf
P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-point algorithms for inverse problems in science and engineering, Springer, 2011, pp. 185–212. doi: https://doi.org/10.1007/978-1-4419-9569-8_10.
J. Nocedal and S. J. Wright, “Numerical optimization,” in Sequential Quadratic Programming, Springer, 1999, pp. 526–573. doi: https://doi.org/10.1007/0-387-22742-3_18.
C. M. Bishop and N. M. Nasrabadi, Pattern recognition and machine learning, 1st ed., vol. 4, no. 4. Springer, 2006. [Online]. Available: https://link.springer.com/book/9780387310732
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M. T. Buber, “Foundations of metric fixed point theory and aspects of constructive fixed points,” The University of Iowa, 1994.
M. Elghandouri, “Approximate Controllability for some Nonlocal Integrodifferential Equations in Banach Spaces,” Sorbonne université, 2024.
L. B. Ćirić, “A generalization of Banach’s contraction principle,” Proc. Am. Math. Soc., vol. 45, no. 2, pp. 267–273, 1974, [Online]. Available: https://www.ams.org/journals/proc/1974-045-02/S0002-9939-1974-0356011-2/
Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” J. Nonlinear Convex Anal., vol. 7, no. 2, p. 289, 2006.
Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete G-metric spaces,” Fixed point theory Appl., vol. 2008, no. 189870, pp. 1–12, 2008, doi: 10.1155/2008/189870.
T. Van An, N. Van Dung, and V. T. Le Hang, “A new approach to fixed point theorems on G-metric spaces,” Topol. Appl., vol. 160, no. 12, pp. 1486–1493, 2013, doi: https://doi.org/10.1016/j.topol.2013.05.027.
D. Vinsensia and Y. Utami, “Fixed Point Theory in Generalized Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms,” Int. J. Basic Appl. Sci., vol. 13, no. 2, pp. 112–122, 2024, doi: https://doi.org/10.35335/ijobas.v13i2.504.
H. Choi, S. Kim, and S. Y. Yang, “Structure for $ g $-Metric Spaces and Related Fixed Point Theorems,” Kyungpook Math. J., vol. 62, no. 4, pp. 773–785, 2022, doi: https://doi.org/10.5666/KMJ.2022.62.4.773.
H. H. Bauschke, “Convex Analysis and Monotone Operator Theory in Hilbert Spaces,” 2011, Springer-Verlag.
H. Iiduka, “Incremental subgradient method for nonsmooth convex optimization with fixed point constraints,” Optim. Methods Softw., vol. 31, no. 5, pp. 931–951, 2016, doi: https://doi.org/10.1080/10556788.2016.1175002.
H. Iiduka, “Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping,” Math. Program., vol. 149, no. 1, pp. 131–165, 2015, doi: https://doi.org/10.1007/s10107-013-0741-1.
H. Lu, R. M. Freund, and Y. Nesterov, “Relatively smooth convex optimization by first-order methods, and applications,” SIAM J. Optim., vol. 28, no. 1, pp. 333–354, 2018, doi: https://doi.org/10.1137/16M1099546.
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L. Bottou, F. E. Curtis, and J. Nocedal, “Optimization methods for large-scale machine learning,” SIAM Rev., vol. 60, no. 2, pp. 223–311, 2018, doi: https://doi.org/10.1137/16M1080173.
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P. Jain and P. Kar, “Non-convex optimization for machine learning,” Found. Trends® Mach. Learn., vol. 10, no. 3–4, pp. 142–363, 2017, doi: http://dx.doi.org/10.1561/2200000058.
K. Bian and R. Priyadarshi, “Machine learning optimization techniques: a Survey, classification, challenges, and Future Research Issues,” Arch. Comput. Methods Eng., vol. 31, no. 7, pp. 4209–4233, 2024, doi: https://doi.org/10.1007/s11831-024-10110-w.
Y. Censor and S. A. Zenios, Parallel optimization: Theory, algorithms, and applications. Oxford University Press, 1997.
B. T. Polyak, “Introduction to optimization,” 1987.
P. Zhou, D. Lin, C. Lu, M. Rosu, and D. M. Ionel, “An Adaptive Fixed-Point Iteration Algorithm for Finite-Element Analysis with Magnetic Hysteresis Materials,” IEEE Trans. Magn., vol. 53, no. 10, pp. 5–6, 2017, doi: 10.1109/TMAG.2017.2712572.
D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in ICLR, 2017, pp. 1–15. doi: https://doi.org/10.48550/arXiv.1412.6980.
Y. Nesterov, “A method for solving the convex programming problem with convergence rate O (1/k2),” in Dokl akad nauk Sssr, 1983, p. 543. [Online]. Available: https://cir.nii.ac.jp/crid/1370862715914709505
W. Kirk and N. Shahzad, Fixed point theory in distance spaces. Springer, 2014. doi: https://doi.org/10.1007/978-3-319-10927-5.
A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy sets Syst., vol. 64, no. 3, pp. 395–399, 1994, doi: https://doi.org/10.1016/0165-0114(94)90162-7.
J. A. Jiddah, H. Işık, M. Alansari, and M. S. Shagari, “A Study of Fixed Point Results in G-Metric Space via New Contractions with Applications,” in Recent Developments in Fixed-Point Theory: Theoretical Foundations and Real-World Applications, Springer, 2024, pp. 241–263. doi: https://doi.org/10.1007/978-981-99-9546-2_10.
R. P. Agarwal, M. Meehan, and D. O’regan, Fixed point theory and applications, vol. 141. Cambridge university press, 2001.
V. François-Lavet, P. Henderson, R. Islam, M. G. Bellemare, and J. Pineau, “An introduction to deep reinforcement learning,” Found. Trends® Mach. Learn., vol. 11, no. 3–4, pp. 219–354, 2018, doi: http://dx.doi.org/10.1561/2200000071.
A. Hossain, F. A. Khan, and Q. H. Khan, “A relation-theoretic metrical fixed point theorem for rational type contraction mapping with an application,” Axioms, vol. 10, no. 4, p. 316, 2021, doi: https://doi.org/10.3390/axioms10040316.
T. Domínguez Benavides, S. M. Moshtaghioun, and A. Sadeghi Hafshejani, “Fixed points for several classes of mappings in Variable Lebesgue Spaces,” Optimization, vol. 70, no. 5–6, pp. 911–927, 2021, doi: https://doi.org/10.1080/02331934.2019.1711086.
I. K. Argyros, The theory and applications of iteration methods. CRC Press, 2022. doi: https://doi.org/10.1201/9781003128915.
S. Raj, C. K. Shiva, B. Vedik, S. Mahapatra, and V. Mukherjee, “A novel chaotic chimp sine cosine algorithm Part-I: For solving optimization problem,” Chaos, Solitons & Fractals, vol. 173, no. 3, p. 113672, 2023, doi: https://doi.org/10.1016/j.chaos.2023.113672.
P. Debnath, N. Konwar, and S. Radenović, Metric fixed point theory, 1st ed. Springer, 2021. doi: https://doi.org/10.1007/978-981-16-4896-0.
V. Berinde and M. Păcurar, “Kannan’s fixed point approximation for solving split feasibility and variational inequality problems,” J. Comput. Appl. Math., vol. 386, no. 4, p. 113217, 2021, doi: https://doi.org/10.1016/j.cam.2020.113217.
M. A. Albreem, A. H. Al Habbash, A. M. Abu-Hudrouss, and S. S. Ikki, “Overview of precoding techniques for massive MIMO,” Ieee Access, vol. 9, no. 11, pp. 60764–60801, 2021, doi: https://doi.org/10.1109/ACCESS.2021.3073325.
J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables. SIAM, 2000. [Online]. Available: https://epubs.siam.org/doi/pdf/10.1137/1.9780898719468.bm
Richard L. Burden and J. D. Faires, Numerical Analysis (9th ed.), 9th ed. Boston, USA: Brooks/Cole, Cengage Learning, 2015. [Online]. Available: https://faculty.ksu.edu.sa/sites/default/files/numerical_analysis_9th.pdf
S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, 2004.
V. Berinde and F. Takens, Iterative approximation of fixed points, 2nd ed., vol. 1912. Romania: Springer, 2007. doi: https://doi.org/10.1007/978-3-540-72234-2.
K. Goebel and W. A. Kirk, Topics in metric fixed point theory, 1st ed., no. 28. Cambridge university press, 1990.
F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proc. Natl. Acad. Sci., vol. 54, no. 4, pp. 1041–1044, 1965, doi: https://doi.org/10.1073/pnas.54.4.1041.
H.-K. Xu, “Iterative algorithms for nonlinear operators,” J. London Math. Soc., vol. 66, no. 1, pp. 240–256, 2002, doi: https://doi.org/10.1112/S0024610702003332.
R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM J. Control Optim., vol. 14, no. 5, pp. 877–898, 1976, doi: https://doi.org/10.1137/0314056.
N. Parikh and S. Boyd, “Proximal algorithms,” Found. trends® Optim., vol. 1, no. 3, pp. 127–239, 2014, doi: http://dx.doi.org/10.1561/2400000003.
H. H. Bauschke, P. L. Combettes, H. H. Bauschke, and P. L. Combettes, “Correction to: convex analysis and monotone operator theory in Hilbert spaces,” in Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, 2017, pp. 1283–1306. doi: https://doi.org/10.1007/978-3-319-48311-5_31.
J. Eckstein and D. P. Bertsekas, “On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Math. Program., vol. 55, no. 4, pp. 293–318, 1992, doi: https://doi.org/10.1007/BF01581204.
S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundam. Math., vol. 3, no. 1, pp. 133–181, 1922.
K. Goebel and W. A. Kirk, “Fixed point theory in metric spaces,” in Topics in Fixed Point Theory, Cambridge University Press, Cambridge, 1990, pp. 101–158. doi: https://doi.org/10.1007/978-3-319-01586-6_4.
E. K. Ryu and S. Boyd, “Primer on monotone operator methods,” in Appl. comput. math, vol. 15, no. 1, 2016, pp. 3–43. [Online]. Available: https://web.stanford.edu/~boyd/papers/pdf/monotone_primer.pdf
P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-point algorithms for inverse problems in science and engineering, Springer, 2011, pp. 185–212. doi: https://doi.org/10.1007/978-1-4419-9569-8_10.
J. Nocedal and S. J. Wright, “Numerical optimization,” in Sequential Quadratic Programming, Springer, 1999, pp. 526–573. doi: https://doi.org/10.1007/0-387-22742-3_18.
C. M. Bishop and N. M. Nasrabadi, Pattern recognition and machine learning, 1st ed., vol. 4, no. 4. Springer, 2006. [Online]. Available: https://link.springer.com/book/9780387310732
I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio, “Deep learning,” Healthc. Inform. Res., vol. 1, no. 2, pp. 1–4, 2016, doi: https://doi.org/10.4258/hir.2016.22.4.351.
M. ABDEHAMID, “Fixed Point Theorems Involving Control Functions in Metric Spaces and Fuzzy Metric Spaces,” Faculté des Sciences et des Techniques, Béni Mellal-Doctorat ou Doctorat …, 2021.
M. T. Buber, “Foundations of metric fixed point theory and aspects of constructive fixed points,” The University of Iowa, 1994.
M. Elghandouri, “Approximate Controllability for some Nonlocal Integrodifferential Equations in Banach Spaces,” Sorbonne université, 2024.
L. B. Ćirić, “A generalization of Banach’s contraction principle,” Proc. Am. Math. Soc., vol. 45, no. 2, pp. 267–273, 1974, [Online]. Available: https://www.ams.org/journals/proc/1974-045-02/S0002-9939-1974-0356011-2/
Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” J. Nonlinear Convex Anal., vol. 7, no. 2, p. 289, 2006.
Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete G-metric spaces,” Fixed point theory Appl., vol. 2008, no. 189870, pp. 1–12, 2008, doi: 10.1155/2008/189870.
T. Van An, N. Van Dung, and V. T. Le Hang, “A new approach to fixed point theorems on G-metric spaces,” Topol. Appl., vol. 160, no. 12, pp. 1486–1493, 2013, doi: https://doi.org/10.1016/j.topol.2013.05.027.
D. Vinsensia and Y. Utami, “Fixed Point Theory in Generalized Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms,” Int. J. Basic Appl. Sci., vol. 13, no. 2, pp. 112–122, 2024, doi: https://doi.org/10.35335/ijobas.v13i2.504.
H. Choi, S. Kim, and S. Y. Yang, “Structure for $ g $-Metric Spaces and Related Fixed Point Theorems,” Kyungpook Math. J., vol. 62, no. 4, pp. 773–785, 2022, doi: https://doi.org/10.5666/KMJ.2022.62.4.773.
H. H. Bauschke, “Convex Analysis and Monotone Operator Theory in Hilbert Spaces,” 2011, Springer-Verlag.
H. Iiduka, “Incremental subgradient method for nonsmooth convex optimization with fixed point constraints,” Optim. Methods Softw., vol. 31, no. 5, pp. 931–951, 2016, doi: https://doi.org/10.1080/10556788.2016.1175002.
H. Iiduka, “Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping,” Math. Program., vol. 149, no. 1, pp. 131–165, 2015, doi: https://doi.org/10.1007/s10107-013-0741-1.
H. Lu, R. M. Freund, and Y. Nesterov, “Relatively smooth convex optimization by first-order methods, and applications,” SIAM J. Optim., vol. 28, no. 1, pp. 333–354, 2018, doi: https://doi.org/10.1137/16M1099546.
S. Lu, I. Tsaknakis, M. Hong, and Y. Chen, “Hybrid block successive approximation for one-sided non-convex min-max problems: algorithms and applications,” IEEE Trans. Signal Process., vol. 68, no. 4, pp. 3676–3691, 2020, doi: https://doi.org/10.1109/TSP.2020.2986363.
L. Bottou, F. E. Curtis, and J. Nocedal, “Optimization methods for large-scale machine learning,” SIAM Rev., vol. 60, no. 2, pp. 223–311, 2018, doi: https://doi.org/10.1137/16M1080173.
C. Jin, R. Ge, P. Netrapalli, S. M. Kakade, and M. I. Jordan, “How to escape saddle points efficiently,” in International conference on machine learning, PMLR, 2017, pp. 1724–1732. [Online]. Available: https://proceedings.mlr.press/v70/jin17a.html
P. Jain and P. Kar, “Non-convex optimization for machine learning,” Found. Trends® Mach. Learn., vol. 10, no. 3–4, pp. 142–363, 2017, doi: http://dx.doi.org/10.1561/2200000058.
K. Bian and R. Priyadarshi, “Machine learning optimization techniques: a Survey, classification, challenges, and Future Research Issues,” Arch. Comput. Methods Eng., vol. 31, no. 7, pp. 4209–4233, 2024, doi: https://doi.org/10.1007/s11831-024-10110-w.
Y. Censor and S. A. Zenios, Parallel optimization: Theory, algorithms, and applications. Oxford University Press, 1997.
B. T. Polyak, “Introduction to optimization,” 1987.
P. Zhou, D. Lin, C. Lu, M. Rosu, and D. M. Ionel, “An Adaptive Fixed-Point Iteration Algorithm for Finite-Element Analysis with Magnetic Hysteresis Materials,” IEEE Trans. Magn., vol. 53, no. 10, pp. 5–6, 2017, doi: 10.1109/TMAG.2017.2712572.
D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in ICLR, 2017, pp. 1–15. doi: https://doi.org/10.48550/arXiv.1412.6980.
Y. Nesterov, “A method for solving the convex programming problem with convergence rate O (1/k2),” in Dokl akad nauk Sssr, 1983, p. 543. [Online]. Available: https://cir.nii.ac.jp/crid/1370862715914709505
W. Kirk and N. Shahzad, Fixed point theory in distance spaces. Springer, 2014. doi: https://doi.org/10.1007/978-3-319-10927-5.
A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy sets Syst., vol. 64, no. 3, pp. 395–399, 1994, doi: https://doi.org/10.1016/0165-0114(94)90162-7.
J. A. Jiddah, H. Işık, M. Alansari, and M. S. Shagari, “A Study of Fixed Point Results in G-Metric Space via New Contractions with Applications,” in Recent Developments in Fixed-Point Theory: Theoretical Foundations and Real-World Applications, Springer, 2024, pp. 241–263. doi: https://doi.org/10.1007/978-981-99-9546-2_10.
R. P. Agarwal, M. Meehan, and D. O’regan, Fixed point theory and applications, vol. 141. Cambridge university press, 2001.
V. François-Lavet, P. Henderson, R. Islam, M. G. Bellemare, and J. Pineau, “An introduction to deep reinforcement learning,” Found. Trends® Mach. Learn., vol. 11, no. 3–4, pp. 219–354, 2018, doi: http://dx.doi.org/10.1561/2200000071.
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