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A globally convergent matrix-free method for constrained equations and its linear convergence rate. (English) Zbl 07022283
Summary: A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factor $$\gamma$$ is attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.

##### MSC:
 65-XX Numerical analysis 90-XX Operations research, mathematical programming
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##### References:
 [1] Zeidler, E., Nonlinear Functional Analysis and Its Applications, (1990), Springer [2] Wood, A. J.; Wollenberg, B. F., Power Generation, Operation, and Control, (1996), New York, NY, USA: John Wiley and Sons, New York, NY, USA [3] Meintjes, K.; Morgan, A. P., A methodology for solving chemical equilibrium systems, Applied Mathematics and Computation, 22, 4, 333-361, (1987) · Zbl 0616.65057 [4] Qi, L.; Tong, X. J.; Li, D. H., Active-set projected trust-region algorithm for box-constrained nonsmooth equations, Journal of Optimization Theory and Applications, 120, 3, 601-625, (2004) · Zbl 1140.65331 [5] Ulbrich, M., Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems, SIAM Journal on Optimization, 11, 4, 889-917, (2001) · Zbl 1010.90085 [6] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, (1970), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0241.65046 [7] Wang, C.; Wang, Y.; Xu, C., A projection method for a system of nonlinear monotone equations with convex constraints, Mathematical Methods of Operations Research, 66, 1, 33-46, (2007) · Zbl 1126.90067 [8] Ma, F.; Wang, C., Modified projection method for solving a system of monotone equations with convex constraints, Journal of Applied Mathematics and Computing, 34, 1-2, 47-56, (2010) · Zbl 1225.90128 [9] Yu, Z.; Lin, J.; Sun, J.; Xiao, Y.; Liu, L.; Li, Z., Spectral gradient projection method for monotone nonlinear equations with convex constraints, Applied Numerical Mathematics, 59, 10, 2416-2423, (2009) · Zbl 1183.65056 [10] Zheng, L., A new projection algorithm for solving a system of nonlinear equations with convex constraints, Bulletin of the Korean Mathematical Society, 50, 3, 823-832, (2013) · Zbl 1273.90158 [11] La Cruz, W.; Raydan, M., Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods & Software, 18, 5, 583-599, (2003) · Zbl 1069.65056 [12] Zhang, L.; Zhou, W., Spectral gradient projection method for solving nonlinear monotone equations, Journal of Computational and Applied Mathematics, 196, 2, 478-484, (2006) · Zbl 1128.65034 [13] Cheng, W., A PRP type method for systems of monotone equations, Mathematical and Computer Modelling, 50, 1-2, 15-20, (2009) · Zbl 1185.65088 [14] Zhang, L.; Zhou, W.; Li, D.-H., A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence, IMA Journal of Numerical Analysis, 26, 4, 629-640, (2006) · Zbl 1106.65056 [15] La Cruz, W.; Martínez, J. M.; Raydan, M., Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75, 255, 1429-1448, (2006) · Zbl 1122.65049
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