Two-step inertial Tseng’s extragradient methods for a class of bilevel split variational inequalities

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 17 فروردین 1404   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.90001.1528

نویسندگان

L.H.M Van^{1، 2، 3}؛ T.V. Anh^{* 4}

¹Department of Mathematics and Physics, University of Information Technology, Ho Chi Minh City, Vietnam;

²Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam;

³Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam.

⁴Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam.

چکیده

This work presents a two-step inertial Tseng’s extragradient method with a self-adaptive step size for solving a bilevel split variational inequality problem (BSVIP) in Hilbert spaces. This algorithm only requires two projections per iteration, enhancing its practicality. We establish a strong convergence theorem for the method, showing that it effectively tackles the BSVIP without necessitating prior knowledge of the Lipschitz or strongly monotone constants associated with the mappings. Additionally, the implementation of this method removes the need to compute or estimate the norm of the given operator, a task that can often be challenging in practical situations. We also explore specific cases to demonstrate the versatility of the method. Finally, we present an application of the split minimum norm problem in production and consumption systems and provide several numerical experiments to validate the practical implementability of
the proposed algorithms.

کلیدواژه‌ها

Bilevel split variational inequality problem؛ Bilevel variational inequality problem؛ Split feasibility problem؛ Tseng’s extragradient method

مراجع

[1] Anh, P.K., Anh, T.V., and Muu, L.D. On bilevel split pseudomonotone variational inequality problems with applications, Acta Math. Vietnam, 42 (2017), 413–429.
[2] Anh, P.N., Kim, J.K., and Muu, L.D. An extragradient algorithm for solving bilevel pseudomonotone variational inequalities, J. Glob. Optim., 52 (2012), 627–639.
[3] Anh, T.V. A parallel method for variational inequalities with the multiple-sets split feasibility problem constraints, J. Fixed Point Theory Appl., 19 (2017), 2681–2696.

[4] Anh, T.V. A strongly convergent subgradient extragradient-Halpern method for solving a class of bilevel pseudomonotone variational in-equalities, Vietnam J. Math., 45 (2017), 317–332.
[5] Buong, N. Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces, Numer. Algorithms, 76 (2017), 783–798.
[6] Ceng, L.C., Ansari, Q.H., and Yao, J.C. Some iterative methods for finding fixed points and for solving constrained convex minimization problems, Nonlinear Anal., 74 (2011), 5286–5302.
[7] Ceng, L.C., Ansari, Q.H., and Yao, J.C. An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633–642.
[8] Ceng, L.C., Ansari, Q.H., and Yao, J.C. Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Anal., 75 (2012), 2116–2125.
[9] Ceng, L.C., Coroian, I., Qin, X., and Yao, J.C. A general viscosity implicit iterative algorithm for split variational inclusions with hierarchical variational inequality constraints, Fixed Point Theory, 20 (2019), 469–482.
[10] Ceng, L.C., Liou, Y.C., and Sahu, D.R. Multi-step hybrid steepest-descent methods for split feasibility problems with hierarchical variational inequality problem constraints, J. Nonlinear Sci. Appl., 9 (2016), 4148–4166.
[11] Ceng, L.C., Wong, N.C., and Yao, J.C. Hybrid extragradient methods for finding minimum-norm solutions of split feasibility problems, J. Nonlinear Convex Anal., 16 (2015), 1965–1983.
[12] Censor, Y., Bortfeld, T., Martin, B., et al. A unified approach for inversion problems in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353–2365.
[13] Censor, Y., and Elfving, T. A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239.

[14] Censor, Y., Elfving, T., Kopf, N. and Bortfeld, T. The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Prob., 21 (2005), 2071–2084.
[15] Censor, Y., Gibali, A., and Reich, S. Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301–323.
[16] Censor, Y., and Segal, A. Iterative projection methods in biomedical inverse problems, in: Censor, Y., Jiang, M., and Louis, A.K., eds., Mathematical methods in biomedical imaging and intensity-modulated therapy, Edizioni della Norale, Pisa, Italy, IMRT, (2008) 65–96.
[17] Combettes, P.L. and Hirstoaga, S.A. Equilibrium Programming in Hilbert Spaces, J. Nonlinear Convex Anal., 6 (2005), 117–136.
[18] Cuong, T.L., Anh, T.V., and Van, L.H.M. A self-adaptive step size algorithm for solving variational inequalities with the split feasibility problem with multiple output sets constraints, Numer. Funct. Anal. Optim., 43(2022), 1009–1026.
[19] Daniele, P., Giannessi, F., and Maugeri, A. Equilibrium problems and variational models, Dordrecht, Kluwer Academic, 2003.
[20] Dempe, S. Foundations of bilevel programming, Dordrecht, Kluwer Academic Press, 2002.
[21] Facchinei, F., and Pang, J.S. Finite-dimensional variational inequalities and complementarity problems, Berlin, Springer, 2002.
[22] Giannessi, F., Maugeri, A., and Pardalos, P.M. Equilibrium problems: nonsmooth optimization and variational inequality models, Dordrecht, Kluwer Academic, 2004.
[23] He, S., Zhao, Z., and Luo, B. A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem, Optimization, 64 (2015), 1907–1918.
[24] Huy, P.V., Hien, N.D., and Anh, T.V. A strongly convergent modified Halpern subgradient extragradient method for solving the split variational inequality problem, Vietnam J. Math., 48 (2020), 187–204.

[25] Huy, P.V., Van, L.H.M., Hien, N.D., and Anh, T.V. Modified Tseng’s extragradient methods with self-adaptive step size for solving bilevel split variational inequality problems, Optimization, 71 (2022), 1721–1748.
[26] Iyiola, O. S., and Shehu, Y. Convergence results of two-step inertial proximal point algorithm, Appl. Numer. Math., 182 (2022), 57–75.
[27] Izuchukwu, C., Aphane, M., and Aremu, K. O. Two-step inertial forward–reflected–anchored–backward splitting algorithm for solving monotone inclusion problems, Comput. Appl. Math., 42 (2023), 351.
[28] Kinderlehrer, D., and Stampacchia, G. An introduction to variational inequalities and their applications, New York, Academic,; 1980.
[29] Konnov, I.V. Combined Relaxation Methods for Variational Inequalities, Springer, Berlin (2000).
[30] Korpelevich, G.M. The extragradient method for finding saddle points and other problems, Ekonomikai Matematicheskie Metody, 12 (1976), 747–756.
[31] Maingé, P.E. A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499–1515.
[32] Okeke, C. C., Jolaoso, L. O., and Shehu, Y. Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces, Int. J. Nonlinear Sci. Numer. Simul., 24 (2021), 769–790.
[33] Polyak, B. T. Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys., 4 (1964), 1–17.
[34] Raeisi, M., Zamani Eskandani, G., and Eslamian, M. A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings, Optimization, 67 (2018), 309–327.
[35] Rudin, W. Functiona l Analysis, 2nd ed., McGraw-Hill, New York, 1991.
[36] Shehu, Y. Strong convergence theorem for multiple sets split feasibility problems in Banach spaces, Numerical Funct. Anal. Optim., 37 (2016), 1021–1036.

[37] Solodov, M.V. An explicit descent method for bilevel convex optimization, J. Convex Anal., 14 (2007), 227–237.
[38] Thong, D.V., and Hieu, D.V. A strong convergence of modified subgradient extragradient method for solving bilevel pseudomonotone variational inequality problems, Optimization, 69 (2020), 1313–1334.
[39] Tseng, P. A modified forward–backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431–446.
[40] Uzor, V., Alakoya, T., and Mewomo, O. T. On split monotone variational inclusion problem with multiple output sets with fixed point constraints, Comput. Methods Appl. Math., 23 (2023), 729–749.
[41] Xu, H.K. Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
[42] Yao, Y., Marino, G., and Muglia, L. A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality, Optimization, 63 (2014), 559–569.
[43] Yao, Y., Postolache, M., and Zhu, Z. Gradient methods with selection technique for the multiple-sets split feasibility problem, Optimization, 69 (2020), 269–281.
[44] Zegeye, H., Shahzad, N., and Yao, Y. Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453–471.