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An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

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Abstract

Consider the nonlinear pseudo-monotone equations over a nonempty closed convex set. A spectral conjugate gradient projection method with the inertial factor is proposed for solving the problem under discussion. Following the projection strategy, we prove that the sequence of spectral parameters is bounded. The search direction generated by the algorithm satisfies the sufficient descent condition and possesses trust region property at each iteration. Under some mild conditions, the global convergence of the proposed method is established without the Lipschitz continuity assumption. Under some standard assumptions, we also establish the linear convergence rate of our method. Preliminary numerical results on constrained nonlinear monotone and pseudo-monotone equations demonstrate the efficiency of the proposed method. Furthermore, to highlight its applicability, we extend our method to deal with logistic regression problems.

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The datasets used or analyzed in the study are available from the author upon reasonable request.

References

  1. Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems. vol-1. Springer, Berlin (2003)

  2. Dirkse, S.P., Ferris, M.C.: MCPLIB: a collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5(4), 319–345 (1995)

    Article  Google Scholar 

  3. Halilu, A.S., Majumder, A., Waziri, M.Y., Ahmed. K.: Signal recovery with convex constrained nonlinear monotone equations through conjugate gradient hybrid approach. Math. Comput. Simulation. 187, 520–539 (2021)

  4. Wood, A.J., Wollenberg, B.F.: Power generation, operation, and control. Wiley, New York (1996)

    Google Scholar 

  5. Chorowski, J., Zurada, J.M.: Learning understandable neural networks with nonnegative weight constraints. IEEE Trans. Neural Netw. Learn. Syst. 26(1), 62–69 (2014)

    Article  MathSciNet  Google Scholar 

  6. Liu, P.J., Shao, H., Wang, Y., Wu, X.Y.: A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations with applications. Appl. Numer. Math. 175, 98–107 (2022)

    Article  MathSciNet  Google Scholar 

  7. Ma, G.D., Jin, J.C., Jian, J.B., Yin, J.H., Han, D.L.: A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing. Numer. Algo. 92(3), 1621–1653 (2022)

    Article  MathSciNet  Google Scholar 

  8. Liu, P.J., Shao, H., Yuan, Z.H., Wu, X.Y., Zheng, T.L.: A family of three-term conjugate gradient projection methods with a restart procedure and their relaxed-inertial extensions for the constrained nonlinear pseudo-monotone equations with applications. Numer. Algo. (2023). https://doi.org/10.1007/s11075-023-01527-8

    Article  MathSciNet  Google Scholar 

  9. Gao, P.T., He, C.J.: An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo 55(4), 53 (2018)

    Article  MathSciNet  Google Scholar 

  10. Yin, J.H., Jian, J.B., Jiang, X.Z., Liu, M.X., Wang, L.Z.: A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications. Numer. Algo. 88(1), 389–418 (2021)

    Article  MathSciNet  Google Scholar 

  11. Liu, P.J., Wu, X.Y., Shao, H., Zhang, Y., Cao, S.H.: Three adaptive hybrid derivative-free projection methods for constrained monotone nonlinear equations and their applications. Numer. Linear Algebra Appl. 30(2), e2471 (2023)

    Article  MathSciNet  Google Scholar 

  12. Abubakar, A.B., Kumam, P., Mohammad, H.: A note on the spectral gradient projection method for nonlinear monotone equations with applications. Comput. Appl. Math. 39(2), 1–35 (2020)

    Article  MathSciNet  Google Scholar 

  13. Yin, J.H., Jian, J.B., Jiang, X.Z.: A spectral gradient projection algorithm for convex constrained nonsmooth equations based on an adaptive line search. Math. Numer. Sin. (Chinese) 42(4), 457–471 (2020)

    MathSciNet  Google Scholar 

  14. Yu, Z.S., Lin, J., Sun, J., Xiao, Y.H., Liu, L.Y., Li, Z.H.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59(10), 2416–2423 (2009)

    Article  MathSciNet  Google Scholar 

  15. Zhang, L., Zhou, W.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196(2), 478–484 (2006)

    Article  MathSciNet  Google Scholar 

  16. Amini, K., Faramarzi, P., Bahrami, S.: A spectral conjugate gradient projection algorithm to solve the large-scale system of monotone nonlinear equations with application to compressed sensing. Int. J. Comput. Math. 99(11), 2290–2307 (2022)

    Article  MathSciNet  Google Scholar 

  17. Aji, S., Kumam, P., Awwal, A.M., Yahaya, M.M., Kumam, W.: Two hybrid spectral methods with inertial effect for solving system of nonlinear monotone equations with application in robotics. IEEE Access 9, 30918–30928 (2021)

    Article  Google Scholar 

  18. Birgin, E.G., Martínez, J.M.: A spectral conjugate gradient method for unconstrained optimization. Appl. Math. Optim. 43(2), 117–128 (2001)

    Article  MathSciNet  Google Scholar 

  19. Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, 22, 355–369 (1998)

  20. Liu, J.K., Li, S.J.: Spectral DY-type projection methods for nonlinear monotone system of equations. J. Comput. Math. 33, 341–355 (2015)

    Article  MathSciNet  Google Scholar 

  21. Liu, J.K., Feng, Y.N.: A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algo. 82(1), 245–262 (2019)

    Article  MathSciNet  Google Scholar 

  22. Dai, Z.F., Chen, X.H., Wen, F.H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015)

    MathSciNet  Google Scholar 

  23. Awwal, A.M., Kumam, P., Abubakar, A.B.: Spectral modified Polak-Ribiére-Polyak projection conjugate gradient method for solving monotone systems of nonlinear equations. Appl. Math. Comput. 362, 124514 (2019)

    MathSciNet  Google Scholar 

  24. Koorapetse, M., Kaelo, P., Lekoko, S., Diphofu, T.: A derivative-free RMIL conjugate gradient projection method for convex constrained nonlinear monotone equations with applications in compressive sensing. Appl. Numer. Math. 165, 431–441 (2021)

    Article  MathSciNet  Google Scholar 

  25. Alvarez, F.: On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38(4), 1102–1102 (2000)

    Article  MathSciNet  Google Scholar 

  26. Gao, X., Cai, X.J., Wang, X.Y.: Self-adaptive inertial projection and contraction algorithm for monotone variational inequality. Asia-Pac. J. Oper. Res. 39(2) (2022)

  27. Andrade, J.S., Lopes, J.D.O., Souza, J.C.D.O.: An inertial proximal point method for difference of maximal monotone vector fields in Hadamard manifolds. J. Global Optim. 85(4), 941–968 (2023)

    Article  MathSciNet  Google Scholar 

  28. Chen, C.H., Chan, R.H., Ma, S.Q., Yang, J.F.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8(4), 2239–2267 (2015)

    Article  MathSciNet  Google Scholar 

  29. Ibrahim, A.H., Kumam, P., Abubakar, A.B., Adamu, A.: Accelerated derivative-free method for nonlinear monotone equations with an application. Numer. Linear Algebra Appl. 29(3), e2424 (2022)

    Article  MathSciNet  Google Scholar 

  30. Jian, J.B., Yin, J.H., Tang, C.M., Han, D.L.: A family of inertial derivative-free projection methods for constrained nonlinear pseudo-monotone equations with applications. Comput. Appl. Math. 41(7), 1–21 (2022)

    Article  MathSciNet  Google Scholar 

  31. Yin, J.H., Jian, J.B., Jiang, X.Z., Wu, X.D.: A family of inertial-relaxed DFPM-based algorithms for solving large-scale monotone nonlinear equations with application to sparse signal restoration. J. Comput. Appl. Math. 419, 114674–114674 (2023)

    Article  MathSciNet  Google Scholar 

  32. Zhang, N., Liu, J.K., Zhang, L.Q., Liu, Z.L.: A fast inertial self-adaptive projection based algorithm for solving large-scale nonlinear monotone equations. J. Comput. Appl. Math. 426, 115087–115087 (2023)

    Article  MathSciNet  Google Scholar 

  33. Ibrahimab, A.H., Kumamacd, P., Rapajiće, S., Pappfg, Z., Abubakarbh, A.B.: Approximation methods with inertial term for large-scale nonlinear monotone equations. Appl. Numer. Math. 181, 417–435 (2022)

    Article  MathSciNet  Google Scholar 

  34. Abubakar, A.B., Kumam, P., Ibrahim, A.H.: Inertial derivative-free projection method for nonlinear monotone operator equations with convex constraints. IEEE Access 9, 92157–92167 (2021)

    Article  Google Scholar 

  35. Ibrahim, A.H., Kumam, P., Sun, M., Chaipunya, P., Abubakar, A.B.: Projection method with inertial step for nonlinear equations: application to signal recovery. J. Ind. Manag. Optim. 19(1), 30–55 (2022)

    Article  MathSciNet  Google Scholar 

  36. Liu, P.J., Shao, H., Yuan, Z.H., Zhou, J.H.: A family of inertial-based derivative-free projection methods with a correction step for constrained nonlinear equations and their applications. Numer. Linear Algebra Appl. (2023). https://doi.org/10.1002/nla.2533

    Article  Google Scholar 

  37. Noinakorn, S., Ibrahim, A.H., Abubakar, A.B., Pakkaranang, N.: A three-term inertial derivative-free projection method for convex constrained monotone equations. Nonlinear Funct. Anal. Appl. 26(4), 839–853 (2021)

    Google Scholar 

  38. Ibrahim, A.H., Kumam, P., Abubakar, A.B., Abubakar, J.: A method with inertial extrapolation step for convex constrained monotone equations. J. Inequal. Appl. 2021(1), 1–25 (2021)

    Article  MathSciNet  Google Scholar 

  39. Aji, S., Kumam, P., Awwal, A.M., Yahaya, M.M., Bakoji, A.M.: A new inertial-based method for solving pseudomonotone operator equations with application. Comput. Appl, Math (2023)

    Book  Google Scholar 

  40. Li, Q., Li, D.H.: A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 31, 1625–1635 (2011)

    Article  MathSciNet  Google Scholar 

  41. Jin, X.B., Zhang, X.Y., Huang, K., Geng, G.G.: Stochastic conjugate gradient algorithm with variance reduction. IEEE Trans. Neural Netw. Learn. Syst. 30(5), 1360–1369 (2018)

    Article  MathSciNet  Google Scholar 

  42. Aji, S., Kumam, P., Awwal, A.M., Yahaya, M.M., Sitthithakerngkiet, K.: An efficient DY-type spectral conjugate gradient method for system of nonlinear monotone equations with application in signal recovery. AIMS Math. 6(8), 8078–8106 (2021)

    Article  MathSciNet  Google Scholar 

  43. Yu, G.H., Niu, S.Z., Ma, J.H.: Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. J. Ind. Manag. Optim. 9(1), 117–129 (2013)

    Article  MathSciNet  Google Scholar 

  44. Zhou, W.J., Li, D.H.: Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 25(1), 89–96 (2007)

    MathSciNet  Google Scholar 

  45. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  Google Scholar 

  46. Papp, Z., Rapají, S.: FR type methods for systems of large-scale nonlinear monotone equations. Appl. Math. Comput. 269, 816–823 (2015)

    MathSciNet  Google Scholar 

  47. Shehu, Y., Dong, Q.L., Jiang, D.: Single projection method for pseudo-monotone variational inequality in Hilbert spaces. Optimization 68(1), 385–409 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the two anonymous referees for their very professional comments and quite useful suggestions, which greatly helped us to improve the original version of this paper.

Funding

This work was supported by the National Natural Science Foundation of China (12171106), the Natural Science Foundation of Guangxi Province (2023GXNSFBA026029), the Middle-aged and Young Teachers’ Basic Ability Promotion Project of Guangxi Province (2023KY0168), Research Project of Guangxi Minzu University (2022KJQD03), and the Postgraduate Research & Innovation Program of Inner Mongolia Autonomous Region, China (B20231033Z).

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Wenli Liu

  2. School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning, 530006, China

    Jinbao Jian & Jianghua Yin

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  1. Wenli Liu
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  2. Jinbao Jian
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All authors read and approved the final manuscript. WL is mainly responsible for algorithm design and theoretical analysis; JJ mainly contributes to algorithm design; JY mainly contributes to theoretical analysis and numerical experiments.

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Correspondence to Jinbao Jian.

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Liu, W., Jian, J. & Yin, J. An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations. Numer Algor (2024). https://doi.org/10.1007/s11075-023-01736-1

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