Thuật toán quán tính tự thích nghi giải bài toán bất đẳng thức biến phân tách áp dụng trong mạng giao thông
Nội dung chính của bài viết
Tóm tắt
Bài báo nghiên cứu bài toán bất đẳng thức biến phân tách trong không gian Hilbert thực. Chúng tôi đề xuất một thuật toán lặp mới kết hợp phương pháp tách tiến-lùi của Tseng với kỹ thuật quán tính và cỡ bước tự thích nghi. Phương pháp chỉ yêu cầu tính đơn điệu và liên tục Lipschitz của các toán tử, không cần giả thiết đồng bức. Điểm đặc biệt là kích thước bước được cập nhật tự động qua các phép tính đơn giản, không cần biết trước chuẩn toán tử hay hằng số Lipschitz. Sự hội tụ mạnh của dãy lặp được chứng minh dưới các điều kiện thông thường. Cuối cùng, chúng tôi áp dụng thuật toán vào bài toán cân bằng mạng giao thông và minh họa hiệu quả của phương pháp qua các thử nghiệm số.
Từ khóa
Bài toán bất đẳng thức biến phân tách, cỡ bước tự thích nghi, toán tử đơn điệu, cân bằng mạng giao thông
Chi tiết bài viết
Tài liệu tham khảo
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[2] Alvarez, F. and Attouch, H. An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping. Set-Valued Analysis, 9(1–2):3–11, 2001. doi: 10.1023/A:1011253113155.
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[12] 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. doi: 10.37193/CJM.2023.03.09.
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[15] Ly, L. X., Nguyen, T. T. T. and Nguyen, Q. A. Relaxed two-step inertial method for solving a class of split variational inequality problems with application to traffic analysis. Optimization, 75(6):1303–1331, 2025. doi: 10.1080/02331934.2025.2459191.
[16] Mewomo, O. T., Uzor, V. A. and Gibali, A. A strongly convergent algorithm for solving split equality problems beyond monotonicity. Computational and Applied Mathematics, 43(6):326, 2024. doi:10.1007/s40314-024-02829-w.
[17] Nguyen, T. T. T. and Tran, T. T. Strong convergence of one-step inertial algorithm for a class of bilevel variational inequalities. Optimization, 75(5):1101–1133, 2026. doi: 10.1080/02331934.2024.2448737.
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[19] Pham, V. H., Nguyen, D. H. and Tran, V. A. A strongly convergent modified Halpern subgradient extragradient method for solving the split variational inequality problem. Vietnam Journal of Mathematics, 48 (1):187–204, 2020. ISSN 2305-2228. doi: 10.1007/s10013-019-00378-y.
[20] Polyak, B. T. Some methods of speeding up the convergence of iteration methods. USSR Computational Mathematics and Mathematical Physics, 4(5):1–17, 1964.
[21] Saejung, S. and Yotkaew, P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 75(2):742–750, 2012. ISSN 0362-546X. doi: 10.1016/j.na.2011.09.005.
[22] Sahu, D. R., Yao, J. C., Verma, M. and Shukla, K. K. Convergence rate analysis of proximal gradient methods with applications to composite minimization problems. Optimization, 70(1):75–100, 2021. doi: 10.1080/02331934.2019.1702040.
[23] Singh, S., Gibali, A. and Qin, X. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal
of Industrial and Management Optimization, 18(1):593–611, 2022. doi:10.3934/jimo.2020170.
[24] Takahashi, W. Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, 2000.
[25] Tan, K. K. and Xu, H. K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications, 178(2):301–308, 1993. ISSN 0022-247X. doi:10.1006/jmaa.1993.1309.
[26] Tseng, P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 38(2): 431–446, 2000. doi: 10.1137/S0363012998338806.
[2] Alvarez, F. and Attouch, H. An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping. Set-Valued Analysis, 9(1–2):3–11, 2001. doi: 10.1023/A:1011253113155.
[3] Batra, C., Chugh, R., Kumar, R., Suwais, K., Almanasra, S.and Mlaiki, N. Strong convergence of split equality variational inequal-
ity, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application. Journal of Inequalities and Applications, 2024(1): 40, 2024. doi: 10.1186/s13660-024-03118-0.
[4] Bauschke, H. H. and Combettes, P. L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Cham, 2 edition, 2017.doi: 10.1007/978-3-319-48311-5.
[5] Byrne, C. A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20(1):103–120, 2004. doi: 10.1088/0266-5611/20/1/006.
[6] Censor, Y. and Elfving, T. A multiprojection algorithm using Bregman projections in a product space. Numerical Algorithms, 8(2):221–239, 1994. doi: 10.1007/BF02142692.
[7] Censor, Y., Gibali, A. and Reich, S. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbertspace. Optimization Methods and Software, 26(4–5):827–845, 2011. doi:10.1080/10556788.2010.551536.
[8] Censor, Y., Gibali, A. and Reich, S. Algorithms for the split variational inequality problem. Numerical Algorithms, 59(2):301–323, 2012.doi: 10.1007/s11075-011-9490-5.
[9] Daniele, P., Maugeri, A. and Oettli, W. Time-dependent traffic equilibria. Journal of Optimization Theory and Applications, 103(3):543–555, 1999. doi: 10.1023/A:1021779823196.
[10] Hieu, D., Anh, P. K. and Muu, L. D. Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl., 66(1):75–96, 1 2017. doi: https://doi.org/10.1007/s10589-016-9857-6.
[11] Izuchukwu, C., Mebawondu, A. A. and Mewomo, O. T. A new method for solving split variational inequality problems without co-coerciveness. Journal of Fixed Point Theory and Applications, 22:98, 2020. doi: 10.1007/s11784-020-00834-0.
[12] 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. doi: 10.37193/CJM.2023.03.09.
[13] Konnov, I. V. Equilibrium Models and Variational Inequalities, volume 210 of Mathematics in Science and Engineering. Elsevier, Amsterdam, 2007. ISBN 978-0-444-53030-1.
[14] Korpelevich, G. M. The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody, 12(4):747–756,1976.
[15] Ly, L. X., Nguyen, T. T. T. and Nguyen, Q. A. Relaxed two-step inertial method for solving a class of split variational inequality problems with application to traffic analysis. Optimization, 75(6):1303–1331, 2025. doi: 10.1080/02331934.2025.2459191.
[16] Mewomo, O. T., Uzor, V. A. and Gibali, A. A strongly convergent algorithm for solving split equality problems beyond monotonicity. Computational and Applied Mathematics, 43(6):326, 2024. doi:10.1007/s40314-024-02829-w.
[17] Nguyen, T. T. T. and Tran, T. T. Strong convergence of one-step inertial algorithm for a class of bilevel variational inequalities. Optimization, 75(5):1101–1133, 2026. doi: 10.1080/02331934.2024.2448737.
[18] Ogwo, G. N., Izuchukwu, C. and Mewomo, O. T. Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity. Numerical Algorithms, 88(3):1419–1456, 2021. doi: 10.1007/s11075-021-01081-1.
[19] Pham, V. H., Nguyen, D. H. and Tran, V. A. A strongly convergent modified Halpern subgradient extragradient method for solving the split variational inequality problem. Vietnam Journal of Mathematics, 48 (1):187–204, 2020. ISSN 2305-2228. doi: 10.1007/s10013-019-00378-y.
[20] Polyak, B. T. Some methods of speeding up the convergence of iteration methods. USSR Computational Mathematics and Mathematical Physics, 4(5):1–17, 1964.
[21] Saejung, S. and Yotkaew, P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 75(2):742–750, 2012. ISSN 0362-546X. doi: 10.1016/j.na.2011.09.005.
[22] Sahu, D. R., Yao, J. C., Verma, M. and Shukla, K. K. Convergence rate analysis of proximal gradient methods with applications to composite minimization problems. Optimization, 70(1):75–100, 2021. doi: 10.1080/02331934.2019.1702040.
[23] Singh, S., Gibali, A. and Qin, X. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal
of Industrial and Management Optimization, 18(1):593–611, 2022. doi:10.3934/jimo.2020170.
[24] Takahashi, W. Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, 2000.
[25] Tan, K. K. and Xu, H. K. Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications, 178(2):301–308, 1993. ISSN 0022-247X. doi:10.1006/jmaa.1993.1309.
[26] Tseng, P. A modified forward-backward splitting method for maximal monotone mappings. SIAM Journal on Control and Optimization, 38(2): 431–446, 2000. doi: 10.1137/S0363012998338806.