Monoton kapsama problemini çözmek için alternatif eylemsiz ileri-geri-ileri ayırma algoritması ve uygulamaları

Öz

Maksimum monoton bir operatör ile tek değerli, aynı zamanda Lipschitz sürekli olan monoton bir operatörün toplamının sıfırlarını bulmak amacıyla tasarlanmış alternatif eylemsiz ileri-geri-ileri algoritmasını sunuyoruz. Bu çalışma, Tseng'in ileri-geri-ileri algoritmasını alternatif atalet etkilerini ekleyerek genişletmeyi amaçlamaktadır. Ardından, geliştirilmiş algoritmamızı konveks minimizasyon problemlerini ele almak için uyguluyoruz. Ana konular arasında monoton kapsama problemi, ileri-geri-ileri algoritması, alternatif eylemsiz yöntemi ve konveks minimizasyon problemleri yer almaktadır. Son olarak, önerdiğimiz yaklaşımın görüntü iyileştirme uygulamasını inceleyerek etkinliğini ve uygulanabilirliğini vurguluyoruz.

Anahtar Kelimeler

• Monoton Kapsama Problemi
• İleri-geri-ileri algoritması
• Alternatif Atalet Metot
• Konveks minimizasyon problemi
• görüntü iyileştirme problemi

An alternating inertial forward-backward-forward algorithm for solving monotone inclusion problem and its applications

Öz

We present an alternating inertial forward-backward-forward algorithm designed to find the zeros of the sum of a maximally monotone operator and a single-valued monotone operator that is also Lipschitz continuous. This study aims to extend Tseng’s forward-backward-forward algorithm by incorporating alternating inertial effects. We then apply our enhanced algorithm to address convex minimization problems. Key topics include the monotone inclusion problem, forward-backward-forward algorithm, the alternating inertial method, and convex minimization problems. Lastly, we explore the application of our proposed approach in image restoration, emphasizing its effectiveness and adaptability.

Anahtar Kelimeler

• Monotone inclusion problem
• forward-backward-forward algorithm
• alternating inertial method
• convex minimization problem
• image restoration problem.

Kaynakça

1. Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society, 73(4), 591-597, (1967).
2. Combettes, P.L., and Wajs, V.R., Signal recovery by proximal forward-backward splitting. Multiscale modeling and simulation 4(4), 1168-1200, (2005).
3. Daubechies, I., Defrise, M., and De Mol, C., An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics: Journal Issued by the Courant Institute of Mathematical Sciences 57(11), 1413-1457, (2004).
4. Duchi, J., and Singer, Y., Efficient online and batch learning using forward backward splitting. The Journal of Machine Learning Research, 10, 2899-2934, (2009).
5. Padcharoen, A., Kitkuan, D., Kumam, and W., Kumam, P., Tseng methods with inertial for solving inclusion problems and application to image deblurring and image recovery problems. Computational and Mathematical Methods 3(3), 1088, (2021).
6. Altiparmak, E., and Karahan, I., A new preconditioning algorithm for finding a zero of the sum of two monotone operators and its application to image restoration problems. International Journal of Computer Mathematics, 99(12), 2482-2498, (2022).
7. Altiparmak, E., and Karahan, I., A modified preconditioning algorithm for solving monotone inclusion problem and application to image restoration problem. Scientific Bulletin-University Politehnica of Bucharest A, 84, 81-92, (2022).
8. Ungchittrakool, K., Cho, Y. J., Plubtieng, S., and Thammasiri, P., Accelerated Mann-type algorithm via two-step inertial points for solving a fixed point problem of a nonexpansive mapping and application to image restoration problems. Numerical Computations: Theory and Algorithms NUMTA 2023, 208, (2023).

9. Jolaoso, L. O., Sunthrayuth, P., Cholamjiak, P., and Cho, Y. J., Inertial projection and contraction methods for solving variational inequalities with applications to image restoration problems. Carpathian Journal of Mathematics, 39(3), 683-704, (2023).
10. Altıparmak, E., Jolaoso, L. O., Karahan, I., and Rehman, H. U., Pre-conditioning CQ algorithm for solving the split feasibility problem and its application to image restoration problem. Optimization, 1-19, (2024).
11. Altiparmak, E., and Karahan, I., A modified inertial viscosity algorithm for an infinite family of nonexpansive mappings and its application to image restoration. Journal of Industrial and Management Optimization, 20(2), 453-477, (2024).
12. Jolaoso, L. O., Bai, J., and Shehu, Y., New fast proximal point algorithms for monotone inclusion problems with applications to image recovery. Optimization, 1-26, (2024).
13. Mungkala, C., Padcharoen, A., and Akkasriworn, N., Convergence of proximal gradient method with alternated inertial step for minimization problem. Advances Fixed Point Theory, 14, Article-ID 35, (2024).
14. Suantai, S., Cholamjiak, P., Inkrong, P., and Kesornprom, S., A fast contraction algorithm using two inertial extrapolations for variational inclusion problem and data classification. Carpathian Journal of Mathematics, 40(3), 737-752, (2024).
15. Suantai, S., Cholamjiak, P., Inkrong, P., and Kesornprom, S., Modified iterative schemes with two inertia and linesearch rule for split variational inclusion and applications to image deblurring and diabetes prediction. Carpathian Journal of Mathematics, 40(2), 459-476, (2024).
16. Jolaoso, L. O., Shehu, Y., and Xu, H. K., New accelerated splitting algorithm for monotone inclusion problems. Optimization, 74(3), 781-810, (2025).
17. Lions, P.L., and Mercier, B. Splitting algorithms for the sum of two nonlinear operators. SIAM Journal on Numerical Analysis, 16, 964-979, (1979).
18. Tseng, P. A., Modified forward-backward splitting method for maximal monotone mappings. SIAM Journal Control Optimization, 38(2), 431-446, (2000).
19. Bot, R. I., Sedlmayer, M., and Vuong, P. T. A relaxed inertial forward-backward-forward algorithm for solving monotone inclusions with application to GANs. Journal of Machine Learning Research, 24(8), 1-37, (2023).
20. Boţ, R. I., and Csetnek, E. R., An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numerical Algorithms, 71, 519-540, (2016).
21. Mu, Z., and Peng, Y., A note on the inertial proximal point method. Statistics, Optimization and Information Computing, 3(3), 241-248, (2015).
22. Iutzeler, F., and Hendrickx, J.M., A generic online acceleration scheme for optimization algorithms via relaxation and inertia. Optimization Methods and Software 34(2), 383-405, (2019).
23. Iutzeler, F., and Malick, J., On the proximal gradient algorithm with alternated inertia. Journal of Optimization Theory and Applications, 176(3), 688-710 (2018).
24. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin, (2011).
25. Takahashi, W., Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, 2009.