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Convergence-Accelerated Fixed-Time Dynamical Methods for Absolute Value Equations

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Abstract

Two new accelerated fixed-time stable dynamic systems are proposed for solving absolute value equations (AVEs): \(Ax-|x|-b=0\). Under some mild conditions, the equilibrium point of the proposed dynamic systems is completely equivalent to the solution of the AVEs under consideration. Meanwhile, we have introduced a new relatively tighter global error bound for the AVEs. Leveraging this finding, we have separately established the globally fixed-time stability of the proposed methods, along with providing the conservative settling-time for each method. Compared with some existing state-of-the-art dynamical methods, preliminary numerical experiments show the effectiveness of our methods in solving the AVEs.

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References

  1. Abdallah, L., Haddou, M., Migot, T.: Solving absolute value equation using complementarity and smoothing functions. J. Comput. Appl. Math. 327, 196–207 (2018)

    MathSciNet  Google Scholar 

  2. Bai, Z.Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17(6), 917–933 (2010)

    MathSciNet  Google Scholar 

  3. Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control. Optim. 38(3), 751–766 (2000)

    MathSciNet  Google Scholar 

  4. Bian, W., Xue, X.P.: Subgradient-based neural networks for nonsmooth nonconvex optimization problems. IEEE Trans. Neural Netw. 20(6), 1024–1038 (2009)

    Google Scholar 

  5. Caccetta, L., Qu, B., Zhou, G.L.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2011)

    MathSciNet  Google Scholar 

  6. Chen, B.T., Harker, P.T.: A continuation method for monotone variational inequalities. Math. Program. 69(1–3), 237–253 (1995)

    MathSciNet  Google Scholar 

  7. Chen, C.R., Yang, Y.N., Yu, D.M., Han, D.R.: An inverse-free dynamical system for solving the absolute value equations. Appl. Numer. Math. 168, 170–181 (2021)

    MathSciNet  Google Scholar 

  8. Chen, C.R., Yu, D.M., Han, D.R.: Exact and inexact Douglas–Rachford splitting methods for solving large-scale sparse absolute value equations. IMA J. Numer. Anal. 43(2), 1036–1060 (2023)

    MathSciNet  Google Scholar 

  9. Chen, F., Ren, W.: Sign projected gradient flow: a continuous-time approach to convex optimization with linear equality constraints. Automatica 120, 109156 (2020)

    MathSciNet  Google Scholar 

  10. Chen, X.J., Qi, L.Q., Sun, D.F.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67(222), 519–540 (1998)

    MathSciNet  Google Scholar 

  11. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Society for Industrial and Applied Mathematics, Philadelphia (2009)

    Google Scholar 

  12. Dai, H., Jia, J.P., Yan, L., Fang, X.P., Chen, W.S.: Distributed fixed-time optimization in economic dispatch over directed networks. IEEE Trans. Ind. Inf. 17(5), 3011–3019 (2020)

    Google Scholar 

  13. De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75, 407–439 (1996)

    MathSciNet  Google Scholar 

  14. Drazin, P.G.: Nonlinear Systems. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  15. Edalatpour, V., Hezari, D., Salkuyeh, D.K.: A generalization of the Gauss-Seidel iteration method for solving absolute value equations. Appl. Math. Comput. 293, 156–167 (2017)

    MathSciNet  Google Scholar 

  16. Eshaghnezhad, S., Effati, M., Mansoori, A.: A neurodynamic model to solve nonlinear pseudo-monotone projection equation and its applications. IEEE Trans. Cybern. 47(10), 3050–3062 (2016)

    Google Scholar 

  17. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997)

    MathSciNet  Google Scholar 

  18. Gao, X.B., Wang, J.: Analysis and application of a one-layer neural network for solving horizontal linear complementarity problems. Int. J. Comput. Intell. Syst. 7(4), 724–732 (2014)

    Google Scholar 

  19. Guo, P., Wu, S.L., Li, C.X.: On the SOR-like iteration method for solving absolute value equations. Appl. Math. Lett. 97, 107–113 (2019)

    MathSciNet  Google Scholar 

  20. Han, D.R.: Inexact operator splitting methods with selfadaptive strategy for variational inequality problems. J. Optim. Theory Appl. 132(2), 227–243 (2007)

    MathSciNet  Google Scholar 

  21. Han, X.: A new single-layer inverse-free fixed-time dynamical system for absolute value equations. https://optimization-online.org/wp-content/uploads/2023/06/IEAVEsonline620-1.pdf

  22. Han, X., He, X., Ju, X.X., Chen, J.W.: Unified single-layer inverse-free neurodynamic network for solving absolute value equations. IEEE Trans. Circuits Syst. II Express Briefs 71(3), 1166–1170 (2024)

    Google Scholar 

  23. Han, X., He, X., Chen, J.W., Ju, X.X.: Unified single-layer inverse-free neurodynamic network for solving absolute value equations. J. Nonlinear Var. Anal. 7(6), 909–923 (2023)

    Google Scholar 

  24. Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15(2), 593–615 (2005)

    MathSciNet  Google Scholar 

  25. He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86(1), 199–217 (1999)

    MathSciNet  Google Scholar 

  26. Iqbal, J., Iqbal, A., Arif, M.: Levenberg–Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134–138 (2015)

    MathSciNet  Google Scholar 

  27. Ju, X.X., Hu, D.Z., Li, C.D., He, X., Feng, G.: A novel fixed-time converging neurodynamic approach to mixed variational inequalities and applications. IEEE Trans. Cybern. 52(12), 12942–12953 (2021)

    Google Scholar 

  28. Ju, X.X., Li, C.D., Han, X., He, X.: Neurodynamic network for absolute value equations: a fixed-time convergence technique. IEEE Trans. Circuits Syst. II Express Briefs 69(3), 1807–1811 (2021)

    Google Scholar 

  29. Ju, X.X., Yang, X.S., Feng, G., Che, H.J.: Neurodynamic optimization approaches with finite/fixed-time convergence for absolute value equations. Neural Netw. 165, 971–981 (2023)

    Google Scholar 

  30. Ke, Y.F., Ma, C.F.: SOR-like iteration method for solving absolute value equations. Appl. Math. Comput. 311, 195–202 (2017)

    MathSciNet  Google Scholar 

  31. Ketabchi, S., Moosaei, H.: Minimum norm solution to the absolute value equation in the convex case. J. Optim. Theory Appl. 154, 1080–1087 (2012)

    MathSciNet  Google Scholar 

  32. Li, X.H., Yu, D.M., Yang, Y.N., Han, D.R., Chen, C.R.: A new fixed-time dynamical system for absolute value equations. Numer. Math. Theory Methods Appl. 16, 622–633 (2022)

    MathSciNet  Google Scholar 

  33. Luo, Z.Q., Tseng, P.: Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem. SIAM J. Optim. 2(1), 43–54 (1992)

    MathSciNet  Google Scholar 

  34. Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419(2–3), 359–367 (2006)

    MathSciNet  Google Scholar 

  35. Mansoori, A., Erfanian, M.: A dynamic model to solve the absolute value equations. J. Comput. Appl. Math. 333, 28–35 (2018)

    MathSciNet  Google Scholar 

  36. Mansoori, A., Eshaghnezhad, M., Effati, S.: An efficient neural network model for solving the absolute value equations. IEEE Trans. Circuits Syst. II Express Briefs 65(3), 391–395 (2017)

    Google Scholar 

  37. Monteiro, R.D.C., Pang, J.S., Wang, T.: A positive algorithm for the nonlinear complementarity problem. SIAM J. Optim. 5(1), 129–148 (1995)

    MathSciNet  Google Scholar 

  38. Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2011)

    MathSciNet  Google Scholar 

  39. Qi, L.Q., Sun, D.F., Zhou, G.L.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Program. 87, 1–35 (2000)

    MathSciNet  Google Scholar 

  40. Romero, O., Benosman, M.: Finite-time convergence in continuous-time optimization. In: H.D. III, Singh, A. (eds.) Proceedings of the 37th International Conference on Machine Learning, vol. 119, pp. 8200–8209. PMLR (2020)

  41. Saheya, B., Nguyen, C.T., Chen, J.S.: Neural network based on systematically generated smoothing functions for absolute value equation. J. Appl. Math. Comput. 61, 533–558 (2019)

    MathSciNet  Google Scholar 

  42. Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191–2202 (2014)

    MathSciNet  Google Scholar 

  43. Shi, X.L., Yu, X.H., Cao, J.D., Wen, G.H.: Continuous distributed algorithms for solving linear equations in finite time. Automatica 113, 108755 (2020)

    MathSciNet  Google Scholar 

  44. Vuong, P.T., Strodiot, J.J.: A dynamical system for strongly pseudo-monotone equilibrium problems. J. Optim. Theory Appl. 185(3), 767–784 (2020)

    MathSciNet  Google Scholar 

  45. Xiu, N.H., Zhang, J.Z.: Global projection-type error bounds for general variational inequalities. J. Optim. Theory Appl. 112(1), 213–228 (2002)

    MathSciNet  Google Scholar 

  46. Xia, Y.S., Leung, H., Wang, J.: A projection neural network and its application to constrained optimization problems. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49(4), 447–458 (2002)

    MathSciNet  Google Scholar 

  47. Yu, D.M., Chen, C.R., Han, D.R.: A modified fixed point iteration method for solving the system of absolute value equations. Optimization 71(3), 449–461 (2022)

    MathSciNet  Google Scholar 

  48. Yu, D.M., Zhang, G.H., Chen, C.R., Han, D.R.: The neural network models with delays for solving absolute value equations. Neurocomputing 589, 127707 (2024)

    Google Scholar 

  49. Zamani, M., Hladík, M.: Error bounds and a condition number for the absolute value equations. Math. Program. 198(1), 85–113 (2023)

    MathSciNet  Google Scholar 

  50. Zeng, X.L., Liang, S., Hong, Y.G., Chen, J.: Distributed computation of linear matrix equations: an optimization perspective. IEEE Trans. Autom. Control 64(5), 1858–1873 (2018)

    MathSciNet  Google Scholar 

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Acknowledgements

Thank you to all the reviewers and editors for their valuable comments and assistance with this manuscript. This work was funded partly by National Key R &D Program of China (2021YFA1003600, 2021YFA1003601), and Key Program of National Natural Science of China (12331011) and National Natural Science Foundation of China (12071398), the Starting Research Fund Project of Xiangtan University (21QDZ50), and Hunan Provincial Innovation Foundation for Postgraduate (CX20230621).

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Authors and Affiliations

  1. School of Automation and Electronic Information, Xiangtan University, Hunan, China

    Xu Zhang

  2. School of Mathematics and Computational Science, Xiangtan University, Hunan, China

    Cailian Li, Longcheng Zhang, Yaling Hu & Zheng Peng

  3. The Key Laboratory of Credible Intelligent Navigation and Positioning of Hunan, Hunan, China

    Xu Zhang

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  1. Xu Zhang
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  2. Cailian Li
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Correspondence to Zheng Peng.

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Communicated by Nobuo Yamashita.

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Zhang, X., Li, C., Zhang, L. et al. Convergence-Accelerated Fixed-Time Dynamical Methods for Absolute Value Equations. J Optim Theory Appl 203, 600–628 (2024). https://doi.org/10.1007/s10957-024-02525-z

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