Smoothing Levenberg-Marquardt algorithm for solving non-Lipschitz absolute value equations

• Autorzy: Yilmaz Nurullah , Kayacan Aysegul

Identyfikatory

DOI

10.1515/jaa-2022-1025

• Języki publikacji: EN

Abstrakty

EN

In this study, we concentrate on solving the problem of non-Lipschitz absolute value equations (NAVE). A new Bezier curve based smoothing technique is introduced and a new Levenberg-Marquardt type algorithm is developed depending on the smoothing technique. The numerical performance of the algorithm is analysed by considering some well-known and randomly generated test problems. Finally, the comparison with other methods is illustrated to demonstrate the efficiency of the proposed algorithm.

Słowa kluczowe

EN

non-Lipschitz absolute value equations; smoothing techniques; Levenberg-Marquardt algorithm

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2023
• Tom: Vol. 29, nr 2
• Strony: 277--286
• Opis fizyczny: Bibliogr. 51 poz., wykr.

Twórcy

autor

Yilmaz Nurullah

• Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

• https://orcid.org/0000-0001-6429-7518

autor

Kayacan Aysegul

• Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

• https://orcid.org/0000-0002-6285-5771

Bibliografia

• [1] L. Abdallah, M. Haddou and T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math. 327 (2018), 196-207.
• [2] J. H. Alcantara, J.-S. Chen and M. K. Tam, Method of alternating projections for the general absolute value equation, J. Fixed Point Theory Appl. 25 (2023), no. 1, Paper No. 39.
• [3] R. Ali, K. Pan and A. Ali, Two new iteration methods with optimal parameters for solving absolute value equations, Int. J. Appl. Comput. Math. 8 (2022), no. 3, Paper No. 123.
• [4] A. M. Bagirov, B. Ordin, G. Ozturk and A. E. Xavier, An incremental clustering algorithm based on hyperbolic smoothing, Comput. Optim. Appl. 61 (2015), no. 1, 219-241.
• [5] A. Ben-Tal and M. Teboulle, A smoothing technique for nondifferentiable optimization problems, in: Optimization (Varetz 1988), Lecture Notes in Math. 1405, Springer, Berlin (1989), 1-11.
• [6] D. P. Bertsekas, Nondifferentiable optimization via approximation, Math. Program. Stud. 3 (1975), 1-25.
• [7] L. Caccetta, B. Qu and G. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl. 48 (2011), no. 1, 45-58.
• [8] C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl. 5 (1996), no. 2, 97-138.
• [9] C. Chen, Y. Yang, D. Yu and D. Han, An inverse-free dynamical system for solving the absolute value equations, Appl. Numer. Math. 168 (2021), 170-181.
• [10] X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program. 134 (2012), no. 1, 71-99.
• [11] S. Du, L. Zhang, C. Chen and L. Qi, Tensor absolute value equations, Sci. China Math. 61 (2018), no. 9, 1695-1710.
• [12] G. E. Farin, Curves and Surfaces for CADG: A Partical Guide, Morgan Kaufmann, San Fransico, 2002.
• [13] A. Goli, E. B. Tirkolaee and G. W. Weber, An integration of neural network and shuffled frog-leaping algorithm for CNC machining monitoring, Found. Comput. Decis. Sci. 46 (2021), 27-42.
• [14] M. Graczyk-Kucharska, M. Szafranski, S. Gutmen, M. Golinski, M. Spychla, G. W. Weber, Z. Wlodarczak, S. Kuter and A. Ozmen, Modeling for human resources management by data mining, analytics and artificial intelligence in the logistics departments, in: Smart and Sustainable Supply Chain and Logistics - Trends, Challenges, Methods and Best Practices, Springer, Cham (2020), 291-303.
• [15] C. Grossmann, Smoothing techniques for exact penalty methods, in: A Panorama of Mathematics: Pure and Applied, Contemp. Math. 658, American Mathematical Society, Providence (2016), 249-265.
• [16] F. Hashemi and S. Ketabchi, Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations, Numer. Algebra Control Optim. 10 (2020), no. 1, 13-21.
• [17] S.-L. Hu and Z.-H. Huang, A note on absolute value equations, Optim. Lett. 4 (2010), no. 3, 417-424.
• [18] J. Iqbal, A. Iqbal and M. Arif, Levenberg-Marquardt method for solving systems of absolute value equations, J. Comput. Appl. Math. 282 (2015), 134-138.
• [19] X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, J. Ind. Manag. Optim. 9 (2013), no. 4, 789-798.
• [20] Y. Ke, The new iteration algorithm for absolute value equation, Appl. Math. Lett. 99 (2020), Article ID 105990.
• [21] Y.-F. Ke and C.-F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput. 311 (2017), 195-202.
• [22] S. Ketabchi and H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, J. Optim. Theory Appl. 154 (2012), no. 3, 1080-1087.
• [23] E. Kürüm, K. Yildirak and G.-W. Weber, A classification problem of credit risk rating investigated and solved by optimisation of the ROC curve, CEJOR Cent. Eur. J. Oper. Res. 20 (2012), no. 3, 529-557.
• [24] S. Kuter, Z. Akyurek and G. W. Weber, Retrieval of fractional snow covered area from MODIS data by multivariate adaptive regression splines, Remote Sens. Environ. 205 (2018), 236-252.
• [25] C.-X. Li and S.-L. Wu, A shift splitting iteration method for generalized absolute value equations, Comput. Methods Appl. Math. 21 (2021), no. 4, 863-872.
• [26] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl. 36 (2007), no. 1, 43-53.
• [27] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett. 3 (2009), no. 1, 101-108.
• [28] O. L. Mangasarian, A hybrid algorithm for solving the absolute value equation, Optim. Lett. 9 (2015), no. 7, 1469-1474.
• [29] O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra Appl. 419 (2006), no. 2-3, 359-367.
• [30] A. Mansoori and M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math. 333 (2018), 28-35.
• [31] X.-H. Miao, J. Yang and S. Hu, A generalized Newton method for absolute value equations associated with circular cones, Appl. Math. Comput. 269 (2015), 155-168.
• [32] X.-H. Miao, K. Yao, C.-Y. Yang and J.-S. Chen, Levenberg-Marquardt method for absolute value equation associated with second-order cone, Numer. Algebra Control Optim. 12 (2022), no. 1, 47-61.
• [33] C. T. Nguyen, B. Saheya, Y.-L. Chang and J.-S. Chen, Unified smoothing functions for absolute value equation associated with second-order cone, Appl. Numer. Math. 135 (2019), 206-227.
• [34] O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl. 44 (2009), no. 3, 363-372.
• [35] H.-D. Qi and L.-Z. Liao, A smoothing Newton method for general nonlinear complementarity problems, Comput. Optim. Appl. 17 (2000), no. 2, 231-253.
• [36] J. Rohn, A theorem of the alternatives for the equation Ax + B|x| = b, Linear Multilinear Algebra 52 (2004), no. 6, 421-426.
• [37] B. Saheya, C. T. Nguyen and J.-S. Chen, Neural network based on systematically generated smoothing functions for absolute value equation, J. Appl. Math. Comput. 61 (2019), no. 1-2, 533-558.
• [38] B. Saheya, C.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, J. Appl. Math. Comput. 56 (2018), no. 1-2, 131-149.
• [39] A. Sahiner, N. Yilmaz and S. A. Ibrahem, Smoothing approximations to non-smooth functions, J. Multidisciplinary Model. Optim. 1 (2018), no. 2, 69-74.
• [40] A. Sahiner, N. Yilmaz, G. Kapusuz and G. Ozkardas, Bezier curve based smoothing penalty function for constrained optimization, J. Multidiscip. Model. Optim. 3 (2021), 70-79.
• [41] N. N. Shams and F. P. A. Beik, An improvement on a class of fixed point iterative methods for solving absolute value equations, Comput. Methods Appl. Math. 22 (2022), no. 3, 663-673.
• [42] J. Tang and J. Zhou, A quadratically convergent descent method for the absolute value equation Ax + B|x| = b, Oper. Res. Lett. 47 (2019), no. 4, 229-234.
• [43] A. Wang, Y. Cao and J.-X. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl. 181 (2019), no. 1, 216-230.
• [44] F. Wang, Z. Yu and C. Gao, A smoothing neural network algorithm for absolute value equations, Engineering 7 (2015), 567-576.
• [45] C. Wu, J. Zhan, Y. Lu and J.-S. Chen, Signal reconstruction by conjugate gradient algorithm based on smoothing l1-norm, Calcolo 56 (2019), no. 4, Paper No. 42.
• [46] A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog. 43 (2010), 731-737.
• [47] N. Yilmaz and A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numer. Funct. Anal. Optim. 40 (2019), no. 5, 513-534.
• [48] N. Yilmaz and A. Sahiner, On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Control Optim. 10 (2020), no. 3, 317-330.
• [49] N. Yilmaz and A. Sahiner, Smoothing techniques in solving non-Lipschitz absolute value equations, Int. J. Comput. Math. (2023), DOI 10.1080/00207160.2022.2163388.
• [50] D. Yu, C. Chen and D. Han, A modified fixed point iteration method for solving the system of absolute value equations, Optimization 71 (2022), no. 3, 449-461.
• [51] I. Zang, A smoothing-out technique for min-max optimization, Math. Programming 19 (1980), no. 1, 61-77.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).