The Hager–Zhang conjugate gradient algorithm for large-scale nonlinear equations

References

• T. Ahmed and D. Storey, Efficient hybrid conjugate gradient techniques, J. Optim. Theory Appl. 64 (1990), pp. 379–394. doi: 10.1007/BF00939455
   Web of Science ®Google Scholar
• A. Al-Baali, Descent property and global convergence of the Flecher–Reeves method with inexact line search, IMA J. Numer. Anal. 5 (1985), pp. 121–124. doi: 10.1093/imanum/5.1.121
   Web of Science ®Google Scholar
• N. Andrei, Another hybrid conjugate gradient algorithm for unconstrained optimization, Numer. Algorithms 47 (2008), pp. 143–156. doi: 10.1007/s11075-007-9152-9
   Web of Science ®Google Scholar
• I. Bongartz, A.R. Conn, N.I.M. Gould and P.L. Toint, CUTE: constrained and unconstrained testing environments, ACM Trans. Math. Software 21 (1995), pp. 123–160. doi: 10.1145/200979.201043
   Web of Science ®Google Scholar
• A. Bouaricha and R.B. Schnabel, Tensor methods for large sparse systems of nonlinear equations, Math. Program. 82 (1998), pp. 377–400.
   Web of Science ®Google Scholar
• S. Buhmiler, N. Krejić and Z. Lužanin, Practical quasi-Newton algorithms for singular nonlinear systems, Numer. Algorithms 55 (2010), pp. 481–502. doi: 10.1007/s11075-010-9367-z
   Web of Science ®Google Scholar
• W. Burmeister, Die Konvergenzordnung des Fletcher–Powell Algorithmus, Z. Angew. Math. Mech. 53 (1973), pp. 693–699. doi: 10.1002/zamm.19730531007
   Web of Science ®Google Scholar
• W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Model. 50 (2009), pp. 15–20. doi: 10.1016/j.mcm.2009.04.007
   Web of Science ®Google Scholar
• A. Cohen, Rate of convergence of several conjugate gradient algorithms, SIAM J. Numer. Anal. 9 (1972), pp. 248–259. doi: 10.1137/0709024
   Web of Science ®Google Scholar
• Y.H. Dai and L.Z. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim. 43 (2001), pp. 87–101. doi: 10.1007/s002450010019
   Web of Science ®Google Scholar
• Y. Dai and Y. Yuan, Nonlinear Conjugate Gradient Methods, Shanghai Scientific and Technical Publishers, Shanghai, 1998.
   Google Scholar
• Y. Dai and Y. Yuan, A nonlinear conjugate gradient with a strong global convergence properties, SIAM J. Optim. 10 (2000), pp. 177–182. doi: 10.1137/S1052623497318992
   Web of Science ®Google Scholar
• Z. Dai, X. Chen and F. Wen, A modified Perry's conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations, Appl. Math. Comput. 270 (2015), pp. 378–386.
   Web of Science ®Google Scholar
• Z. Dai, D. Li and F. Wen, Worse-case conditional value-at-risk for asymmetrically distributed asset scenarios returns, J. Comput. Anal. Appl. 20 (2016), pp. 237–251.
   Web of Science ®Google Scholar
• J.W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), pp. 10–26. doi: 10.1137/0704002
   Google Scholar
• E.D. Dolan and J.J. Moré, Benchmarking optimization software with performance profiles, Math. Program. 91 (2002), pp. 201–213. doi: 10.1007/s101070100263
   Web of Science ®Google Scholar
• G. Fasano, F. Lampariello and M. Sciandrone, A truncated nonmonotone Gauss–Newton method for large-scale nonlinear least-squares problems, Comput. Optim. Appl. 34 (2006), pp. 343–358.
   Google Scholar
• R. Fletcher, Practical Methods of Optimization, 2nd ed., John Wiley and Sons, Chichester, 1987.
   Google Scholar
• R. Fletcher and C.M. Reeves, Function minimization by conjugate gradients, Comput. J. 7 (1964), pp. 149–154. doi: 10.1093/comjnl/7.2.149
   Web of Science ®Google Scholar
• J.C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim. 2 (1992), pp. 21–42. doi: 10.1137/0802003
   Web of Science ®Google Scholar
• L. Grippo and M. Sciandrone, Nonmonotone derivative-free methods for nonlinear equations, Comput. Optim. Appl. 37 (2007), pp. 297–328. doi: 10.1007/s10589-007-9028-x
   Web of Science ®Google Scholar
• W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim. 16 (2005), pp. 170–192. doi: 10.1137/030601880
   Web of Science ®Google Scholar
• W. Hager and H. Zhang, Algorithm 851: CGDESCENT, A conjugate gradient method with guaranteed descent, ACM Trans. Math. Software 32 (2006), pp. 113–137. doi: 10.1145/1132973.1132979
   Web of Science ®Google Scholar
• M.R. Hestenes and E. Stiefel, Method of conjugate gradient for solving linear equations, J. Res. Natl. Bur. Stand. 49 (1952), pp. 409–436. doi: 10.6028/jres.049.044
   Google Scholar
• Y.F. Hu and C. Storey, Global convergence result for conjugate method, J. Optim. Theory Appl. 71 (1991), pp. 399–405. doi: 10.1007/BF00939927
   Web of Science ®Google Scholar
• C. Kanzow, N. Yamashita and M. Fukushima, Levenberg–Marquardt methods for constrained nonlinear equations with strong local convergence properties, J. Comput. Appl. Math. 172 (2004), pp. 375–397. doi: 10.1016/j.cam.2004.02.013
   Web of Science ®Google Scholar
• W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw. 18 (2003), pp. 583–599. doi: 10.1080/10556780310001610493
   Web of Science ®Google Scholar
• W. La Cruz, J.M. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput. 75 (2006), pp. 1429–1448. doi: 10.1090/S0025-5718-06-01840-0
   Web of Science ®Google Scholar
• D. Li and M. Fukushima, A global and superlinear convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations, SIAM J. Numer. Anal. 37 (1999), pp. 152–172. doi: 10.1137/S0036142998335704
   Web of Science ®Google Scholar
• Q. Li and D. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal. 31 (2011), pp. 1625–1635. doi: 10.1093/imanum/drq015
   Web of Science ®Google Scholar
• D.H. Li and B.S. Tian, N-step quadratic convergence of the MPRP method with a restart strategy, J. Comput. Appl. Math. 235 (2011), pp. 4978–4990. doi: 10.1016/j.cam.2011.04.026
   Web of Science ®Google Scholar
• D. Li and L. Wang, A modified Fletcher–Reeves-type derivative-free method for symmetric nonlinear equations, Numer. Algebra Control Optim. 1 (2011), pp. 71–82. doi: 10.3934/naco.2011.1.71
   Google Scholar
• D. Li, L. Qi and S. Zhou, Descent directions of quasi-Newton methods for symmetric nonlinear equations, SIAM J. Numer. Anal. 40 (2002), pp. 1763–1774. doi: 10.1137/S003614290139175X
   Web of Science ®Google Scholar
• Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms part 1: theory, J. Appl. Math. Comput. 69 (1992), pp. 17–41.
   Google Scholar
• D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (1963), pp. 431–441. doi: 10.1137/0111030
   Web of Science ®Google Scholar
• E. Polak, The conjugate gradient method in extreme problems, Comput. Math. Math. Phys. 9 (1969), pp. 94–112. doi: 10.1016/0041-5553(69)90035-4
   Google Scholar
• E. Polak and G. Ribière, Note sur la convergence de directions conjugees, Rev. Fran. Inf. Rech. Opérat. 3 (1969), pp. 35–43.
   Google Scholar
• M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim. 7 (1997), pp. 26–33. doi: 10.1137/S1052623494266365
   Web of Science ®Google Scholar
• K. Ritter, On the rate of superlinear convergence of a class of variable metric methods, Numer. Math. 35 (1980), pp. 293–313. doi: 10.1007/BF01396414
   Web of Science ®Google Scholar
• Z.J. Shi, Convergence of line search methods for unconstrained optimization, Appl. Math. Comput. 157 (2004), pp. 393–405.
   Web of Science ®Google Scholar
• M.V. Solodov and B.F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, 1998, pp. 355–369.
   Google Scholar
• P.L. Toint, Numerical solution of large sets of algebraic nonlinear equations, Math. Comput. 173 (1986), pp. 175–189. doi: 10.1090/S0025-5718-1986-0815839-9
   Google Scholar
• X. Tong and L. Qi, On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions, J. Optim. Theory Appl. 123 (2004), pp. 187–211. doi: 10.1023/B:JOTA.0000043997.42194.dc
   Web of Science ®Google Scholar
• C.W. Wang, Y.J. Wang and C.L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res. 66 (2007), pp. 33–46. doi: 10.1007/s00186-006-0140-y
   Web of Science ®Google Scholar
• Z. Wei, S. Yao and L. Liu, The convergence properties of some new conjugate gradient methods, Appl. Math. Comput. 183 (2006), pp. 1341–1350.
   Web of Science ®Google Scholar
• Z. Wei, G. Li and L. Qi, Global convergence of the PRP conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems, Math. Comput. 77 (2008), pp. 2173–2193. doi: 10.1090/S0025-5718-08-02031-0
   Web of Science ®Google Scholar
• F. Wen, Z. He, Z. Dai, and X. Yang, Characteristics of investors' risk preference for stock markets, Econ. Comput. Econ. Cybernet. Stud. Res. 48 (2014), pp. 235–254.
   Web of Science ®Google Scholar
• G. Yu, L. Guan and W. Chen, Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization, Optim. Methods Softw. 23 (2008), pp. 275–293. doi: 10.1080/10556780701661344
   Web of Science ®Google Scholar
• Z.S. Yu, J. Lina, J. Sun, Y.H. Xiao, L. Liu and Z.H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math. 59 (2009), pp. 2416–2423. doi: 10.1016/j.apnum.2009.04.004
   Web of Science ®Google Scholar
• Y.X. Yuan, Analysis on the conjugate gradient method, Optim. Methods Softw. 2 (1993), pp. 19–29. doi: 10.1080/10556789308805532
   Google Scholar
• G. Yuan, Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems, Optim. Lett. 3 (2009), pp. 11–21. doi: 10.1007/s11590-008-0086-5
   Web of Science ®Google Scholar
• Y. Yuan, Subspace methods for large scale nonlinear equations and nonlinear least squares, Optim. Eng. 10 (2009), pp. 207–218. doi: 10.1007/s11081-008-9064-0
   Web of Science ®Google Scholar
• G. Yuan and X. Lu, A new backtracking inexact BFGS method for symmetric nonlinear equations, Comput. Math. Appl. 55 (2008), pp. 116–129. doi: 10.1016/j.camwa.2006.12.081
   Web of Science ®Google Scholar
• G. Yuan and X. Lu, A modified PRP conjugate gradient method, Ann. Oper. Res. 166 (2009), pp. 73–90. doi: 10.1007/s10479-008-0420-4
   Web of Science ®Google Scholar
• G. Yuan and S. Yao, A BFGS algorithm for solving symmetric nonlinear equations, Optimization 62 (2013), pp. 82–95. doi: 10.1080/02331934.2011.564621
   Web of Science ®Google Scholar
• G. Yuan and M. Zhang, A modified Hestenes–Stiefel conjugate gradient algorithm for large-scale optimization, Numer. Funct. Anal. Optim. 34 (2013), pp. 914–937. doi: 10.1080/01630563.2013.777350
   Web of Science ®Google Scholar
• G. Yuan and M. Zhang, A three-terms Polak–Ribière–Polyak conjugate gradient algorithm for large-scale nonlinear equations, J. Comput. Appl. Math. 286 (2015), pp. 186–195. doi: 10.1016/j.cam.2015.03.014
   Web of Science ®Google Scholar
• G. Yuan, X. Lu and Z. Wei, A conjugate gradient method with descent direction for unconstrained optimization, J. Comput. Appl. Math. 233 (2009), pp. 519–530. doi: 10.1016/j.cam.2009.08.001
   Web of Science ®Google Scholar
• G. Yuan, X. Lu and Z. Wei, BFGS trust-region method for symmetric nonlinear equations, J. Comput. Appl. Math. 230 (2009), pp. 44–58. doi: 10.1016/j.cam.2008.10.062
   Web of Science ®Google Scholar
• G. Yuan, Z. Wei and X. Lu, A BFGS trust-region method for nonlinear equations, Computing 92 (2011), pp. 317–333. doi: 10.1007/s00607-011-0146-z
   Web of Science ®Google Scholar
• G. Yuan, Z. Wei and S. Lu, Limited memory BFGS method with backtracking for symmetric nonlinear equations, Math. Comput. Model. 54 (2011), pp. 367–377. doi: 10.1016/j.mcm.2011.02.021
   Web of Science ®Google Scholar
• G. Yuan, Z. Wei and Q. Zhao, A modified Polak–Ribière–Polyak conjugate gradient algorithm for large-scale optimization problems, IIE Trans. 46 (2014), pp. 397–413. doi: 10.1080/0740817X.2012.726757
   Web of Science ®Google Scholar
• J. Zhang and Y. Wang, A new trust region method for nonlinear equations, Math. Methods Oper. Res. 58 (2003), pp. 283–298. doi: 10.1007/s001860300302
   Web of Science ®Google Scholar
• L. Zhang and W.J. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, J. Comput. Appl. Math. 196 (2006), pp. 478–484. doi: 10.1016/j.cam.2005.10.002
   Web of Science ®Google Scholar
• L. Zhang, W. Zhou and D. Li, A descent modified Polak–Ribière–Polyak conjugate method and its global convergence, IMA J. Numer. Anal. 26 (2006), pp. 629–649. doi: 10.1093/imanum/drl016
   Web of Science ®Google Scholar