×

zbMATH — the first resource for mathematics

Two spectral gradient projection methods for constrained equations and their linear convergence rate. (English) Zbl 1310.65066
Summary: Due to its simplicity and numerical efficiency for unconstrained optimization problems, the spectral gradient method has received more and more attention in recent years. In this paper, two spectral gradient projection methods for constrained equations are proposed, which are combinations of the well-known spectral gradient method and the hyperplane projection method. The new methods are not only derivative-free, but also completely matrix-free, and consequently they can be applied to solve large-scale constrained equations. Under the condition that the underlying mapping of the constrained equations is Lipschitz continuous or strongly monotone, we establish the global convergence of the new methods. Compared with the existing gradient methods for solving such problems, the new methods possess a linear convergence rate under some error bound conditions. Furthermore, a relax factor \(\gamma\) is attached in the update step to accelerate convergence. Preliminary numerical results show that they are efficient and promising in practice.

MSC:
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C06 Large-scale problems in mathematical programming
Software:
MCPLIB
PDF BibTeX Cite
Full Text: DOI
References:
[1] Dirkse, SP; Ferris, MC, MCPLIB: a collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5, 319-345, (1995)
[2] Wood, AJ, Wollenberg, BF: Power Generation, Operation, and Control. Wiley, New York (1996)
[3] Meintjes, K; Morgan, AP, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22, 333-361, (1987) · Zbl 0616.65057
[4] Qi, LQ; Tong, XJ; Li, DH, An active-set projected trust region algorithm for box constrained nonsmooth equations, J. Optim. Theory Appl., 120, 601-625, (2004) · Zbl 1140.65331
[5] Ortega, JM, Rheinboldt, WC: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970) · Zbl 0241.65046
[6] Yu, ZS; Lin, J; Sun, J; Xiao, YH; Liu, LY; Li, ZH, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59, 2416-2423, (2009) · Zbl 1183.65056
[7] Liu, SY; Huang, YY; Jiao, HW, Sufficient descent conjugate gradient methods for solving convex constrained nonlinear monotone equations, Abstr. Appl. Anal., 2014, (2014)
[8] Sun, M; Liu, J, Three derivative-free projection methods for large-scale nonlinear equations with convex constraints, J. Appl. Math. Comput., (2014) · Zbl 1348.90630
[9] Barzilai, J; Borwein, JM, Two point stepsize gradient methods, IMA J. Numer. Anal., 8, 141-148, (1988) · Zbl 0638.65055
[10] Birgin, EG; Martinez, JM; Raydan, M, Spectral projected gradient methods: review and perspectives, J. Stat. Softw., 60, 1-21, (2014)
[11] Fletcher, R; Reeves, C, Function minimization by conjugate gradients, Comput. J., 7, 149-154, (1964) · Zbl 0132.11701
[12] Dai, YH; Liao, LZ, \(R\)-linear convergence of the Barzilai and Borwein gradient method, IMA J. Numer. Anal., 22, 1-10, (2002) · Zbl 1002.65069
[13] Wang, CW; Wang, YJ; Xu, CL, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66, 33-46, (2007) · Zbl 1126.90067
[14] Zheng, L, A new projection algorithm for solving a system of nonlinear equations with convex constraints, Bull. Korean Math. Soc., 50, 823-832, (2013) · Zbl 1273.90158
[15] Xiao, YH; Zhu, H, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405, 310-319, (2013) · Zbl 1316.90050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.