Extension of modified Polak-Ribière-Polyak conjugate gradient method to linear equality constraints minimization problems. (English) Zbl 1474.90519

Summary: Combining the Rosen gradient projection method with the two-term Polak-Ribière-Polyak (PRP) conjugate gradient method, we propose a two-term Polak-Ribière-Polyak (PRP) conjugate gradient projection method for solving linear equality constraints optimization problems. The proposed method possesses some attractive properties: (1) search direction generated by the proposed method is a feasible descent direction; consequently the generated iterates are feasible points; (2) the sequences of function are decreasing. Under some mild conditions, we show that it is globally convergent with Armijio-type line search. Preliminary numerical results show that the proposed method is promising.


90C53 Methods of quasi-Newton type
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods


Full Text: DOI


[1] Rosen, J. B., The gradient projection method for nonlinear programming. I. Linear constraints, SIAM Journal on Applied Mathematics, 8, 1, 181-217 (1960) · Zbl 0099.36405
[2] Du, D. Z.; Zhang, X. S., A convergence theorem of Rosen’s gradient projection method, Mathematical Programming, 36, 2, 135-144 (1986) · Zbl 0626.90077
[3] Du, D., Remarks on the convergence of Rosen’s gradient projection method, Acta Mathematicae Applicatae Sinica, 3, 2, 270-279 (1987) · Zbl 0624.65051
[4] Du, D. Z.; Zhang, X. S., Global convergence of Rosen’s gradient projection method, Mathematical Programming, 44, 1, 357-366 (1989) · Zbl 0683.90081
[5] Polak, E.; Ribire, G., Note surla convergence de directions conjuguees, Rev Francaise informat Recherche Operatinelle 3e Annee, 16, 35-43 (1969)
[6] Polyak, B. T., The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9, 4, 94-112 (1969) · Zbl 0229.49023
[7] Cheng, W., A two-term PRP-based descent method, Numerical Functional Analysis and Optimization, 28, 11, 1217-1230 (2007) · Zbl 1138.90028
[8] Hager, W. W.; Zhang, H., A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM Journal on Optimization, 16, 1, 170-192 (2005) · Zbl 1093.90085
[9] Li, G.; Tang, C.; Wei, Z., New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, Journal of Computational and Applied Mathematics, 202, 2, 523-539 (2007) · Zbl 1116.65069
[10] Andrei, N., Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, European Journal of Operational Research, 204, 3, 410-420 (2010) · Zbl 1189.90151
[11] Yu, G.; Guan, L.; Chen, W., Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization, Optimization Methods and Software, 23, 2, 275-293 (2008) · Zbl 1279.90166
[12] Yuan, G., Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems, Optimization Letters, 3, 1, 11-21 (2009) · Zbl 1154.90623
[13] Li, Q.; Li, D.-H., A class of derivative-free methods for large-scale nonlinear monotone equations, IMA Journal of Numerical Analysis, 31, 4, 1625-1635 (2011) · Zbl 1241.65047
[14] Xiao, Y.; Zhu, H., A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, Journal of Mathematical Analysis and Applications, 405, 1, 310-319 (2013) · Zbl 1316.90050
[15] Dai, Z., Two modified HS type conjugate gradient methods for unconstrained optimization problems, Nonlinear Analysis: Theory, Methods & Applications, 74, 3, 927-936 (2011) · Zbl 1203.49049
[16] Chen, Y.-Y.; Du, S.-Q., Nonlinear conjugate gradient methods with Wolfe type line search, Abstract and Applied Analysis, 2013 (2013) · Zbl 1278.90374
[17] Liu, S. Y.; Huang, Y. Y.; Jiao, H. W., Sufficient descent conjugate gradient methods for solving convex constrained nonlinear monotone equations, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.65104
[18] Dai, Z. F.; Li, D. H.; Wen, F. H., Robust conditional value-at-risk optimization for asymmetrically distributed asset returns, Pacific Journal of Optimization, 8, 3, 429-445 (2012) · Zbl 1264.90201
[19] Huang, C.; Peng, C.; Chen, X.; Wen, F., Dynamics analysis of a class of delayed economic model, Abstract and Applied Analysis, 2013 (2013) · Zbl 1273.91314
[20] Huang, C.; Gong, X.; Chen, X.; Wen, F., Measuring and forecasting volatility in Chinese stock market using HAR-CJ-M model, Abstract and Applied Analysis, 2013 (2013) · Zbl 1273.91401
[21] Qin, G.; Huang, C.; Xie, Y.; Wen, F., Asymptotic behavior for third-order quasi-linear differential equations, Advances in Difference Equations, 30, 13, 305-312 (2013)
[22] Huang, C.; Kuang, H.; Chen, X.; Wen, F., An LMI approach for dynamics of switched cellular neural networks with mixed delays, Abstract and Applied Analysis, 2013 (2013) · Zbl 1283.34069
[23] Cui, Q. F.; Wang, Z. G.; Chen, X.; Wen, F., Sufficient conditions for non-Bazilevic functions, Abstract and Applied Analysis, 2013 (2013) · Zbl 1295.30030
[24] Wen, F.; Yang, X., Skewness of return distribution and coefficient of risk premium, Journal of Systems Science & Complexity, 22, 3, 360-371 (2009)
[25] Wen, F.; Liu, Z., A copula-based correlation measure and its application in Chinese stock market, International Journal of Information Technology & Decision Making, 8, 4, 787-801 (2009) · Zbl 1186.91236
[26] Wen, F.; He, Z.; Chen, X., Investors’ risk preference characteristics and conditional skewness, Mathematical Problems in Engineering, 2014 (2014)
[27] Wen, F.; Gong, X.; Chao, Y.; Chen, X., The effects of prior outcomes on risky choice: evidence from the stock market, Mathematical Problems in Engineering, 2014 (2014) · Zbl 1407.91286
[28] Wen, F.; He, Z.; Gong, X.; Liu, A., Investors’ risk preference characteristics based on different reference point, Discrete Dynamics in Nature and Society, 2014 (2014) · Zbl 1422.91380
[29] Dai, Z.; Wen, F., Robust CVaR-based portfolio optimization under a genal a_ne data perturbation uncertainty set, Journal of Computational Analysis and Applications, 16, 1, 93-103 (2014) · Zbl 1401.91514
[30] Martínez, J. M.; Pilotta, E. A.; Raydan, M., Spectral gradient methods for linearly constrained optimization, Journal of Optimization Theory and Applications, 125, 3, 629-651 (2005) · Zbl 1079.90164
[31] Li, C.; Li, D. H., An extension of the Fletcher-Reeves method to linear equality constrained optimization problem, Applied Mathematics and Computation, 219, 23, 10909-10914 (2013) · Zbl 1302.65156
[32] Li, C.; Fang, L.; Cui, X., A feasible fletcher-reeves method to linear equality constrained optimization problem, Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA ’10)
[33] Zhang, L.; Zhou, W. J.; Li, D. H., Global convergence of a modified Fletcher-Reeves conjugate gradient method with Armijo-type line search, Numerische Mathematik, 104, 2, 561-572 (2006) · Zbl 1103.65074
[34] Gould, N. I. M.; Hribar, M. E.; Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM Journal on Scientific Computing, 23, 4, 1376-1395 (2001) · Zbl 0999.65050
[35] Hock, W.; Schittkowski, K., Test examples for nonlinear programming codes. Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems (1981), New York, NY, USA: Springer, New York, NY, USA · Zbl 0393.90072
[36] Asaadi, J., A computational comparison of some non-linear programs, Mathematical Programming, 4, 1, 144-154 (1973) · Zbl 0259.90044
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.