Global Convergence Properties of Nonlinear Conjugate Gradient Methods with Modified Secant Condition
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Conjugate gradient methods are appealing for large scale nonlinear optimization problems. Recently, expecting the fast convergence of the methods, Dai and Liao (2001) used secant condition of quasi-Newton methods. In this paper, we make use of modified secant condition given by Zhang et al. (1999) and Zhang and Xu (2001) and propose a new conjugate gradient method following to Dai and Liao (2001). It is new features that this method takes both available gradient and function value information and achieves a high-order accuracy in approximating the second-order curvature of the objective function. The method is shown to be globally convergent under some assumptions. Numerical results are reported.
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- 1.M. Al-Baali, “Descent property and global convergence of the Fletcher-Reeves method with inexact line search,” IMA Journal of Numerical Analysis, vol. 5, pp. 121-124, 1985.Google Scholar
- 2.Y.H. Dai and L.Z. Liao, “New conjugacy conditions and related nonlinear conjugate gradient methods,” Applied Mathematics and Optimization, vol. 43, pp. 87-101, 2001.Google Scholar
- 3.Y.H. Dai, J.Y. Han, G.H. Liu, D.F. Sun, H.X.Yin, and Y.Yuan, “Convergence properties of nonlinear conjugate gradient methods,” SIAM Journal on Optimization, vol. 10, pp. 345-358, 1999.Google Scholar
- 4.J.C. Gilbert and J. Nocedal, “Global convergence properties of conjugate gradient methods for optimization,” SIAM Journal on Optimization, vol. 2, pp. 21-42, 1992.Google Scholar
- 5.J.J. Moré, B.S. Garbow, and K.E. Hillstrom, “Testing unconstrained optimization software,”ACM Transactions on Mathematical Software, vol. 7, pp. 17-41, 1981.Google Scholar
- 6.J. Nocedal and S.J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer-Verlag: New York, 1999.Google Scholar
- 7.M.J.D. Powell, “Nonconvex minimization calculations and the conjugate gradient method,” in Lecture Notes in Mathematics, no. 1066, Springer-Verlag, Berlin, 1984, pp. 122-141.Google Scholar
- 8.J.Z. Zhang, N.Y. Deng, and L.H. Chen, “New quasi-Newton equation and related methods for unconstrained optimization,” Journal of Optimization Theory and Applications, vol. 102, pp. 147-167, 1999.Google Scholar
- 9.J.Z. Zhang and C.X. Xu, “Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations,” Journal of Computational and Applied Mathematics, vol. 137, pp. 269-278, 2001.Google Scholar
- 10.G. Zoutendijk, “Nonlinear programming, computational methods,” in Integer and Nonlinear Programming, J. Abadie (Ed.), North-Holland: Amsterdam, 1970, pp. 37-86.Google Scholar