A Modified Three-Term Hestenes–Stiefel Conjugate Gradient Method with Guaranteed Descent Property

Conjugate gradient (CG) methods are among the most widely used iterative algorithms for large-scale unconstrained optimization, valued for their simplicity and minimal memory requirements. A central challenge is ensuring guaranteed sufficient descent of the search direction, independent of the line search procedure. This paper proposes a Modified Three-Term Hestenes–Stiefel (MTTHS) conjugate gradient method that enforces the sufficient descent condition exactly, without any line search restriction. The proposed direction incorporates a geometric correction term θₖyₖ into the standard three-term framework, where the scalar θₖ is derived analytically to satisfy gₖ₊₁ᵀdₖ₊₁ = −‖gₖ₊₁‖². Global convergence is established under the strong Wolfe line search conditions via the Zoutendijk convergence criterion. Numerical experiments were based on 23 test problems, each solved at five different dimensions, n ∈ {100, 500, 1000, 5000, 10000}, yielding 115 test cases. The results show that the MTTHS method is robust and computationally efficient

conjugate gradient; Hestenes–Stiefel; three-term; sufficient descent; strong Wolfe line search; global convergence; large-scale optimization

1. Al-Baali, M. (1985). Descent property and global convergence of the Fletcher–Reeves method with inexact line search. IMA Journal of Numerical Analysis, 5(1), 121–124. [Google Scholar] [Crossref]

2. Al-Baali, M., & Grandinetti, L. (2014). On practical modifications of the Hestenes–Stiefel conjugate gradient algorithm. Numerical Algorithms, 67(4), 879–902. [Google Scholar] [Crossref]

3. Andrei, N. (2013). Another conjugate gradient algorithm with guaranteed descent and conjugacy conditions for large-scale unconstrained optimization. Journal of Optimization Theory and Applications, 159(1), 159–182. [Google Scholar] [Crossref]

4. Dai, Y. H., & Liao, L. Z. (2001). New conjugacy conditions and related nonlinear conjugate gradient methods. Applied Mathematics and Optimization, 43(1), 87–101. [Google Scholar] [Crossref]

5. Dai, Y. H., & Yuan, Y. (1999). A nonlinear conjugate gradient method with a strong global convergence property. SIAM Journal on Optimization, 10(1), 177–182. [Google Scholar] [Crossref]

6. Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2), 201–213. [Google Scholar] [Crossref]

7. Fletcher, R., & Reeves, C. M. (1964). Function minimization by conjugate gradients. The Computer Journal, 7(2), 149–154. [Google Scholar] [Crossref]

8. Gould, N. I. M., Orban, D., & Toint, Ph. L. (2015). CUTEst: A constrained and unconstrained testing environment with safe threads. Computational Optimization and Applications, 60(3), 545–557. [Google Scholar] [Crossref]

9. Hager, W. W., & Zhang, H. (2005). A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM Journal on Optimization, 16(1), 170–192. [Google Scholar] [Crossref]

10. Hassan, B. A., Moghrabi, I. A., Ibrahim, A. L., & Jabbar, H. N. (2025). Improved conjugate gradient methods for unconstrained minimization problems and training recurrent neural network. Engineering Reports, 7(2), e70019. [Google Scholar] [Crossref]

11. Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409–436. [Google Scholar] [Crossref]

12. Ibrahim, A. L., Fathi, B. G., & Abdulrazzaq, M. B. (2025). Improving three-term conjugate gradient methods for training artificial neural networks in accurate heart disease prediction. Neural Computing and Applications, 37(16), 10381–10405. [Google Scholar] [Crossref]

13. Kumar, K., Ghosh, D., Yao, J. C., & Zhao, X. (2025). Nonlinear conjugate gradient methods for unconstrained set optimization problems whose objective functions have finite cardinality. Optimization, 74(15), 3839–3878. [Google Scholar] [Crossref]

14. Li, G., Tang, C., & Wei, Z. (2008). New conjugacy condition and related new conjugate gradient methods for unconstrained optimization. Journal of Computational and Applied Mathematics, 202(2), 523–539. [Google Scholar] [Crossref]

15. Liu, Y., & Storey, C. (1991). Efficient generalized conjugate gradient algorithms, part 1: Theory. Journal of Optimization Theory and Applications, 69(1), 129–137. [Google Scholar] [Crossref]

16. Moré, J. J., Garbow, B. S., & Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software, 7(1), 17–41. [Google Scholar] [Crossref]

17. Narushima, Y., Yabe, H., & Ford, J. A. (2011). A three-term conjugate gradient method with sufficient descent property for unconstrained optimization. SIAM Journal on Optimization, 21(1), 212–230. [Google Scholar] [Crossref]

18. Nocedal, J., & Wright, S. J. (2006). Numerical optimization (2nd ed.). Springer. [Google Scholar] [Crossref]

19. Polak, E., & Ribière, G. (1969). Note sur la convergence de méthodes de directions conjuguées. Revue Française d'Informatique et de Recherche Opérationnelle, 3(16), 35–43. [Google Scholar] [Crossref]

20. Polyak, B. T. (1969). The conjugate gradient method in extremal problems. USSR Computational Mathematics and Mathematical Physics, 9(4), 94–112. [Google Scholar] [Crossref]

21. Yahaya, J., & Kumam, P. (2025). New hybrid conjugate gradient algorithm for vector optimization problems. Computational and Applied Mathematics, 44(2), Article 163. [Google Scholar] [Crossref]

22. Yunus, R. B., Zainuddin, N., Daud, H., Kannan, R., Yahaya, M. M., & Al-Yaari, A. (2025). An improved accelerated 3-term conjugate gradient algorithm with second-order Hessian approximation for nonlinear least-squares optimization. Journal of Mathematics and Computer Science, 36(3), 263–274. [Google Scholar] [Crossref]

23. Zhang, L., Zhou, W., & Li, D. H. (2007). A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence. IMA Journal of Numerical Analysis, 26(4), 629–640. [Google Scholar] [Crossref]

24. Zoutendijk, G. (1960). Methods of feasible directions (Vol. 960). Elsevier, Amsterdam. [Google Scholar] [Crossref]

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