Yuan, Gonglin; Lu, Xiwen A modified PRP conjugate gradient method. (English) Zbl 1163.90798 Ann. Oper. Res. 166, 73-90 (2009). Summary: This paper gives a modified PRP method which possesses the global convergence of nonconvex function and the R-linear convergence rate of uniformly convex function. Furthermore, the presented method has sufficiently descent property and characteristic of automatically being in a trust region without carrying out any line search technique. Numerical results indicate that the new method is interesting for the given test problems. Cited in 35 Documents MSC: 90C52 Methods of reduced gradient type Keywords:line search; unconstrained optimization; conjugate gradient method; global convergence; R-linear convergence Software:CUTEr; SifDec; minpack PDF BibTeX XML Cite \textit{G. Yuan} and \textit{X. Lu}, Ann. Oper. Res. 166, 73--90 (2009; Zbl 1163.90798) Full Text: DOI References: [1] Al-baali, A. (1985). Descent property and global convergence of the Fletcher-Reeves method with inexact line search. IMA Journal of Numerical Analysis, 5, 121–124. · Zbl 0578.65063 [2] Armijo, L. (1966). Minimization of functions having Lipschitz conditions partial derivatives. Pacific Journal of Mathematics, 16, 1–3. · Zbl 0202.46105 [3] Chen, X., & Sun, J. (2002). Global convergence of a two-parameter family of conjugate gradient methods without line search. Journal of Computational and Applied Mathematics, 146(1), 37–45. · Zbl 1018.65081 [4] Dai, Y. (2001). Convergence of nonlinear conjugate methods. Journal of Computational Mathematics, 19, 539–549. · Zbl 1008.65039 [5] Dai, Y. (2002). A nonmonotone conjugate gradient algorithm for unconstrained optimization. Journal of Systems Science and Complexity, 15, 139–145. · Zbl 1019.90039 [6] Dai, Y., & Ni, Q. (2003). Testing different conjugate gradient methods for large-scale unconstrained optimization. Journal of Computational Mathematics, 21(3), 311–320. · Zbl 1041.65048 [7] Dai, Y., & Yuan, Y. (1995). Further studies on the Polak-Ribière-Polyak method. Research report ICM-95-040, Institute of Computational Mathematics and Scientific/Engineering computing, Chinese Academy of Sciences. [8] Dai, Y., & Yuan, Y. (1996). Convergence properties of the conjugate descent method. Advances in Mathematics, 6, 552–562. · Zbl 0871.49028 [9] Dai, Y., & Yuan, Y. (2000a). A nonlinear conjugate gradient with a strong global convergence properties. SIAM Journal on Optimization, 10, 177–182. · Zbl 0957.65061 [10] Dai, Y., & Yuan, Y. (2000b). Nonlinear conjugate gradient methods. Shanghai: Science Press of Shanghai. [11] Dai, Y., & Yuan, Y. (2001). An efficient hybrid conjugate gradient method for unconstrained optimization. Annals of Operations Research, 103, 33–47. · Zbl 1007.90065 [12] Dai, Y., Han, J., Liu, G., Sun, D., Yin, H., & Yan, Y. (1999). Convergence properties of the nonlinear conjugate methods. SIAM Journal on Optimization, 2, 345–358. · Zbl 0957.65062 [13] Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91, 201–213. · Zbl 1049.90004 [14] Fletcher, R. (1997). Practical method of optimization, Vol I: Unconstrained optimization (2nd ed.). New York: Wiley. [15] Fletcher, R., & Reeves, C. (1964). Function minimization by conjugate gradients. Computer Journal, 7, 149–154. · Zbl 0132.11701 [16] Gibert, J. C., & Nocedal, J. (1992). Global convergence properties of conjugate gradient methods for optimization. SIAM Journal on Optimization, 2, 21–42. · Zbl 0767.90082 [17] Gould, N. I. M., Orban, D., Toint, P. L., & CUTEr(and SifDec) (2003). A constrained and unconstrained testing environment, revisited. ACM Transactions on Mathematical Software, 29, 373–394. · Zbl 1068.90526 [18] Griewank, A. (1989). On automatic differentiation. In M. Iri & K. Tanabe (Eds.), Mathematical programming: Recent developments and applications (pp. 84–108). Dordrecht: Kluwer Academic. · Zbl 0696.65015 [19] Grippo, L., & Lucidi, S. (1997). A globally convergent version of the Polak-Ribière gradient method. Mathematical Programming, 78, 375–391. · Zbl 0887.90157 [20] Grippo, L., Lampariello, F., & Lucidi, S. (1986). A nonmonotone line search technique for Newton’s method. SIAM Journal on Numerical Analysis, 23, 707–716. · Zbl 0616.65067 [21] Han, J., Liu, G., Sun, D., & Yin, H. (2001). Two fundamental convergence theorems for nonlinear conjugate gradient methods and their applications. Acta Mathematicae Applicatae Sinica, 1, 38–46. · Zbl 0973.65048 [22] Hestenes, M. R., & Stiefel, E. (1952). Method of conjugate gradient for solving linear equations. Journal of Research of National Bureau of Standards, 49, 409–436. · Zbl 0048.09901 [23] Liu, G., & Han, J. (1995). On the global convergence of conjugate gradient methods with inexact line search. Numerical Mathematics. A Journal of Chinese Universities, 2, 147–153. · Zbl 0845.65024 [24] Liu, Y., & Storey, C. (1992). Efficient generalized conjugate gradient algorithms, part 1: theory. Journal of Optimization Theory and Application, 69, 17–41. [25] Liu, G., Han, J., & Yin, H. (1995). Global convergence of the Fletcher-Reeves algorithm with inexact line search. Applied Mathematics. Journal of Chinese Universities Series B, 10, 75–82. · Zbl 0834.90122 [26] McCormick, G. (1977). A modification of Armijo’s step-size rule for negative curvature. Mathematical Programming, 13, 111–115. · Zbl 0367.90100 [27] Moré, J. J., Garbow, B. S., & Hillstrome, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software, 7, 17–41. · Zbl 0454.65049 [28] Nocedal, J. (1992). Theorem of algorithms for unconstrained optimization. In Acta Numerica (pp. 199–242). Cambridge: Cambridge University Press. · Zbl 0766.65051 [29] Nocedal, J. (1995). Conjugate gradient methods and nonlinear optimization. In L. Adams & J. L. Nazareth (Eds.), Linear and nonlinear conjugate gradient related methods (pp. 9–23). Philadelphia: SIAM. · Zbl 0866.65037 [30] Polak, E. (1997). Optimization: Algorithms and consistent approximations. New York: Springer. · Zbl 0899.90148 [31] Polak, E., & Ribière, G. (1969). Note sur la convergence de directions conjugèes. Rev. Francaise Inform. Recherche Operationelle, 16(3), 35–43. · Zbl 0174.48001 [32] Polyak, B. T. (1969). The conjugate gradient method in extreme problems. USSR Computational Mathematics and Mathematical Physics, 9, 94–112. · Zbl 0229.49023 [33] Powell, M. J. D. (1977). Restart procedures for the conjugate gradient method. Mathematical Programming, 12, 241–254. · Zbl 0396.90072 [34] Powell, M. J. D. (1984). Nonconvex minimization calculations and the conjugate gradient method. In Lecture notes in mathematics (vol. 1066, pp. 122–141). Berlin: Springer. · Zbl 0531.65035 [35] Powell, M. J. D. (1986). Convergence properties of algorithms for nonlinear optimization. SIAM Review, 28, 487–500. · Zbl 0624.90091 [36] Qi, H., Han, J., & Liu, G. (1996). A modification of Hestenes-Stiefel conjugate gradient method. Chinese Annals of Mathematics, 3, 177–184. · Zbl 0918.65045 [37] Shi, Z. J. (2004). Convergence of line search methods for unconstrained optimization. Applied Mathematics and Computation, 157, 393–405. · Zbl 1072.65087 [38] Sun, J., & Zhang, J. (2001). Convergence of conjugate gradient methods without line search. Annals of Operations Research, 103, 161–173. · Zbl 1014.90071 [39] Wei, Z., Li, G., & Qi, L. (2006a). New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Applied Mathematics and Computation, 179, 407–430. · Zbl 1106.65055 [40] Wei, Z., Yao, S., & Liu, L. (2006b). The convergence properties of some new conjugate gradient methods. Applied Mathematics and Computation, 183, 1341–1350. · Zbl 1116.65073 [41] Yuan, Y. (1993). Analysis on the conjugate gradient method. Optimization Methods and Software, 2, 19–29. [42] Zhang et al. (2006). A descent modified Polak-Ribière-Polyak conjugate method and its global convergence. IMA Journal on Numerical Analysis, 26, 629–649. · Zbl 1106.65056 [43] Zoutendijk, G. (1970). Nonlinear programming computational methods. In J. Abadie (Ed.), Integer and nonlinear programming (pp. 37–86). Amsterdam: North-Holland. · Zbl 0336.90057 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.