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doi: 10.3934/jimo.2021125
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## A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints

 Department of Mathematics, Hainan University, Haikou 570228, China

* Corresponding author: Yigui Ou

Received  August 2020 Revised  March 2021 Early access August 2021

Fund Project: Supported by NNSF of China (No. 11961018), NSF of Hainan Province (No. 120QN175) and Innovative Project for Postgraduates of Hainan Province (No. Hys2020-107)

Motivated by recent derivative-free projection methods proposed in the literature for solving nonlinear constrained equations, in this paper we propose a unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Under mild conditions, the global convergence and convergence rate of the proposed method are established. In order to verify the feasibility and effectiveness of the model, a practical algorithm is devised and the corresponding numerical experiments are reported, which show that the proposed practical method is efficient and can be applied to solve large-scale nonsmooth equations. Moreover, the proposed practical algorithm is also extended to solve the obstacle problem.

Citation: Yigui Ou, Wenjie Xu. A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021125
##### References:
 [1] A. B. Abubakar, P. Kumam and H. Mohammad, A note on the spectral gradient projection method for nonlinear monotone equations with applications, Comput. Appl. Math., 39 (2020), Paper No. 129, 35 pp. doi: 10.1007/s40314-020-01151-5.  Google Scholar [2] K. Amini and A. Kamandi, A new line search strategy for finding separating hyperplane in projection-based methods, Numer. Algorithms, 70 (2015), 559-570.  doi: 10.1007/s11075-015-9961-1.  Google Scholar [3] A. M. Awwal, P. Kumama and A. B. Abubakar, A modified conjugate gradient method for monotone nonlinear equations with convex constraints, Applied Numerical Mathematics, 145 (2019), 507-520.  doi: 10.1016/j.apnum.2019.05.012.  Google Scholar [4] S. Babaie-Kafaki and Z. Aminifard, Two-parameter scaled memoryless BFGS methods with a nonmonotone choice for the initial step length, Numer. Algorithms, 82 (2019), 1345-1357.  doi: 10.1007/s11075-019-00658-1.  Google Scholar [5] S. C. Billups and K. G. Murty, Complementarity problems, J. Comput. Appl. Math., 124 (2000), 303-318.  doi: 10.1016/S0377-0427(00)00432-5.  Google Scholar [6] E. D. Dolan and J. J. Mor$\acute{e}$, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar [7] P. Gao and C. He, An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints, Calcolo, 55 (2018), Paper No. 53, 17 pp. doi: 10.1007/s10092-018-0291-2.  Google Scholar [8] A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distance for constrained optimization, Optimization, 41 (1997), 257-278.  doi: 10.1080/02331939708844339.  Google Scholar [9] C.-X. Jia and D.-T. Zhu, Projected gradient trust-region method for solving nonlinear systems with convex constraints, Appl. Math. J. Chinese Univ. Ser. B, 26 (2011), 57-69.  doi: 10.1007/s11766-011-1956-7.  Google Scholar [10] C. Kanzow, N. Yamashita and M. Fukushima, Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, J. Comput. Appl. Math., 173 (2005), 321-343.  doi: 10.1016/j.cam.2004.03.015.  Google Scholar [11] M. Koorapetse, P. Kaelo and E. R. Offen, A scaled derivative-free projection method for solving nonlinear monotone equations, Bull. Iranian Math. Soc., 45 (2019), 755-770.  doi: 10.1007/s41980-018-0163-1.  Google Scholar [12] J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, 82 (2019), 245-262.  doi: 10.1007/s11075-018-0603-2.  Google Scholar [13] J. Liu and S. Li, Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.  doi: 10.3934/jimo.2016017.  Google Scholar [14] K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Transactions on Mathematical Software, 16 (1990), 143-151.  doi: 10.1145/78928.78930.  Google Scholar [15] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.   Google Scholar [16] Y. Ou and J. Li, A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints, J. Appl. Math. Comput., 56 (2018), 195-216.  doi: 10.1007/s12190-016-1068-x.  Google Scholar [17] Y. Ou and Y. Liu, Supermemory gradient methods for monotone nonlinear equations with convex constraints, Comput. Appl. Math., 36 (2017), 259-279.  doi: 10.1007/s40314-015-0228-1.  Google Scholar [18] J.-S. Pang, Inexact Newton methods for the nonlinear complementary problem, Math. Programming, 36 (1986), 54-71.  doi: 10.1007/BF02591989.  Google Scholar [19] B. T. Polyak, Introduction to Optimization, Optimization Software Incorporation, Publications Division, New York, NY, USA, 1987.  Google Scholar [20] M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for system of monotone equations, in: M. Fukushima and L.Qi (Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, Dordrecht, (1999), 355–369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar [21] M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar [22] W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. Google Scholar [23] Z. Wan, J. Guo, J. J. Liu and W. Y. Liu, A modified spectral conjugate gradient projectionmethod for signal recovery, Signal Image Video Process, 12 (2018), 1455-1462.   Google Scholar [24] C. Wang, Y. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar [25] X. Y. Wang, S. J. Li and X. P. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145.  doi: 10.1007/s10092-015-0140-5.  Google Scholar [26] A. J. Wood and B. F. Wollenberg, Power Generations, Operations, and Control, Wiley, New York, 1996. Google Scholar [27] Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.  Google Scholar [28] Z. Yu, J. Lin, J. Sun, Y. Xiao, L. Liu and Z. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar [29] Y.-B. Zhao and D. Li, Monotonicity of fixed point and normal mapping associated with variational inequality and is applications, SIAM J. Optim., 11 (2001), 962-973.  doi: 10.1137/S1052623499357957.  Google Scholar [30] L. Zheng, A new projection algorithm for solving a system of nonlinear equations with convex constraints, Bull. Korean Math. Soc., 50 (2013), 823-832.  doi: 10.4134/BKMS.2013.50.3.823.  Google Scholar

show all references

##### References:
 [1] A. B. Abubakar, P. Kumam and H. Mohammad, A note on the spectral gradient projection method for nonlinear monotone equations with applications, Comput. Appl. Math., 39 (2020), Paper No. 129, 35 pp. doi: 10.1007/s40314-020-01151-5.  Google Scholar [2] K. Amini and A. Kamandi, A new line search strategy for finding separating hyperplane in projection-based methods, Numer. Algorithms, 70 (2015), 559-570.  doi: 10.1007/s11075-015-9961-1.  Google Scholar [3] A. M. Awwal, P. Kumama and A. B. Abubakar, A modified conjugate gradient method for monotone nonlinear equations with convex constraints, Applied Numerical Mathematics, 145 (2019), 507-520.  doi: 10.1016/j.apnum.2019.05.012.  Google Scholar [4] S. Babaie-Kafaki and Z. Aminifard, Two-parameter scaled memoryless BFGS methods with a nonmonotone choice for the initial step length, Numer. Algorithms, 82 (2019), 1345-1357.  doi: 10.1007/s11075-019-00658-1.  Google Scholar [5] S. C. Billups and K. G. Murty, Complementarity problems, J. Comput. Appl. Math., 124 (2000), 303-318.  doi: 10.1016/S0377-0427(00)00432-5.  Google Scholar [6] E. D. Dolan and J. J. Mor$\acute{e}$, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar [7] P. Gao and C. He, An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints, Calcolo, 55 (2018), Paper No. 53, 17 pp. doi: 10.1007/s10092-018-0291-2.  Google Scholar [8] A. N. Iusem and M. V. Solodov, Newton-type methods with generalized distance for constrained optimization, Optimization, 41 (1997), 257-278.  doi: 10.1080/02331939708844339.  Google Scholar [9] C.-X. Jia and D.-T. Zhu, Projected gradient trust-region method for solving nonlinear systems with convex constraints, Appl. Math. J. Chinese Univ. Ser. B, 26 (2011), 57-69.  doi: 10.1007/s11766-011-1956-7.  Google Scholar [10] C. Kanzow, N. Yamashita and M. Fukushima, Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, J. Comput. Appl. Math., 173 (2005), 321-343.  doi: 10.1016/j.cam.2004.03.015.  Google Scholar [11] M. Koorapetse, P. Kaelo and E. R. Offen, A scaled derivative-free projection method for solving nonlinear monotone equations, Bull. Iranian Math. Soc., 45 (2019), 755-770.  doi: 10.1007/s41980-018-0163-1.  Google Scholar [12] J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, 82 (2019), 245-262.  doi: 10.1007/s11075-018-0603-2.  Google Scholar [13] J. Liu and S. Li, Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.  doi: 10.3934/jimo.2016017.  Google Scholar [14] K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems, ACM Transactions on Mathematical Software, 16 (1990), 143-151.  doi: 10.1145/78928.78930.  Google Scholar [15] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.   Google Scholar [16] Y. Ou and J. Li, A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints, J. Appl. Math. Comput., 56 (2018), 195-216.  doi: 10.1007/s12190-016-1068-x.  Google Scholar [17] Y. Ou and Y. Liu, Supermemory gradient methods for monotone nonlinear equations with convex constraints, Comput. Appl. Math., 36 (2017), 259-279.  doi: 10.1007/s40314-015-0228-1.  Google Scholar [18] J.-S. Pang, Inexact Newton methods for the nonlinear complementary problem, Math. Programming, 36 (1986), 54-71.  doi: 10.1007/BF02591989.  Google Scholar [19] B. T. Polyak, Introduction to Optimization, Optimization Software Incorporation, Publications Division, New York, NY, USA, 1987.  Google Scholar [20] M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for system of monotone equations, in: M. Fukushima and L.Qi (Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, Dordrecht, (1999), 355–369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar [21] M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar [22] W. Y. Sun and Y. X. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. Google Scholar [23] Z. Wan, J. Guo, J. J. Liu and W. Y. Liu, A modified spectral conjugate gradient projectionmethod for signal recovery, Signal Image Video Process, 12 (2018), 1455-1462.   Google Scholar [24] C. Wang, Y. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar [25] X. Y. Wang, S. J. Li and X. P. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145.  doi: 10.1007/s10092-015-0140-5.  Google Scholar [26] A. J. Wood and B. F. Wollenberg, Power Generations, Operations, and Control, Wiley, New York, 1996. Google Scholar [27] Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.  Google Scholar [28] Z. Yu, J. Lin, J. Sun, Y. Xiao, L. Liu and Z. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar [29] Y.-B. Zhao and D. Li, Monotonicity of fixed point and normal mapping associated with variational inequality and is applications, SIAM J. Optim., 11 (2001), 962-973.  doi: 10.1137/S1052623499357957.  Google Scholar [30] L. Zheng, A new projection algorithm for solving a system of nonlinear equations with convex constraints, Bull. Korean Math. Soc., 50 (2013), 823-832.  doi: 10.4134/BKMS.2013.50.3.823.  Google Scholar
Performance profile for the number of iterations
Performance profile for the number of function evaluations
Performance profile for the CPU time
An elastic string stretched over an obstacle
Numerical test results for the obstale problem
 n Algorithm5.1 OLA (CPU/FN) XZA (CPU/FN) 50 1.664626/9.7218e-06 1.814302/7.1023e-06 4.972539/9.9601e-06 100 10.632101/5.9094e-05 10.647692/8.0831e-05 39.520620/1.3001e-05 500 90.768111/0.0161 167.333069/0.0174 459.419363/0.0279
 n Algorithm5.1 OLA (CPU/FN) XZA (CPU/FN) 50 1.664626/9.7218e-06 1.814302/7.1023e-06 4.972539/9.9601e-06 100 10.632101/5.9094e-05 10.647692/8.0831e-05 39.520620/1.3001e-05 500 90.768111/0.0161 167.333069/0.0174 459.419363/0.0279
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