A Modification of Dai-Yuan's Conjugate Gradient Algorithm for Solving Unconstrained Optimization

Метод спектральных сопряженных градиентов является существенным обобщением метода сопряженных градиентов, а также одним из эффективных численных методов для решения крупномасштабных задач безусловной оптимизации. Мы предложили новый спектральный метод сопряженных градиентов Дай-Юаня для решения нелинейных задач безусловной оптимизации. Глобальная сходимость предложенного метода была достигнута при соответствующих условиях, проведены численные эксперименты на 65 эталонных тестах, показывающие эффективность предложенного метода по сравнению с другими методами, такими как алгоритм AMDYN и некоторыми другими существующими методами, такими как метод Дай-Юаня.

Ключевые слова

неограниченная оптимизация; метод сопряженных градиентов; спектральный сопряженный градиент; достаточный спуск; глобальная конвергенция.

Литература

1. Alhawarat A., Salleh Z. Modification of Nonlinear Conjugate Gradient Method with Weak Wolfe-Powell Line Search. Abstract and Applied Analysis, 2017, vol. 2017, no. 2, pp. 1-6. DOI: 10.1155/2017/7238134
2. Nocedal J., Yuan Ya-Xiang. Analysis of a Self-Scaling Quasi-Newton Method. Mathematical Programming, 1993, vol. 61, no. 1, pp. 19-37. DOI: 10.1007/BF01582136
3. Hestenes M.R., Stiefel E. Methods of Conjugate Gradients for Solving Linear Systems. Journal of Research of the National Bureau of Standards, vol. 49, no. 6, pp. 409-435. DOI: 10.6028/jres.049.044
4. Fletcher R., Reeve C.M., Function Minimization by Conjugate Gradients. The Computer Journal, 1964, vol. 7, no. 2, pp. 149-154. DOI: 10.1093/comjnl/7.2.149
5. Polak E., Ribiere G. Note sur la convergence de m'ethodes de directions conjugu'ees. Mathematical Modelling and Numerical Analysis-Mod'elisation Math'ematique et Analyse Num'erique, 1969, vol. 3, no. 1, pp. 35-43. (in French)
6. Polyak B.T. The Conjugate Gradient Method in Extremal Problems. Computational Mathematics and Mathematical Physics, 1969, vol. 9, no. 4, pp. 94-112. DOI: 10.1016/0041-5553(69)90035-4
7. Dai Yu-Hong, Yuan Yaxiang. A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property. SIAM Journal on Optimization, 1999, vol. 10, no. 1, pp. 177-182.
8. Dai Yu-Hong, Li-Zhi Liao. New Conjugacy Conditions and Related Nonlinear Conjugate Gradient Methods. Applied Mathematics and Optimization, 2001, vol. 43, no. 1, pp. 87-101. DOI: 10.1007/s002450010019
9. Hassan B., Abdullah Z., Jabbar H. A Descent Extension of the Dai-Yuan Conjugate Gradient Technique. Indonesian Journal of Electrical Engineering and Computer Science, 2019, vol. 16, no. 2, pp. 661-668. DOI: 10.11591/ijeecs.v16.i2.pp661-668
10. Jie Guo, Zhong Wan. A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large-Scale Monotone Equations and its Application in Compressed Sensing. Mathematical Problems in Engineering, 2019, vol. 2019, pp. 23-27. DOI: 10.1155/2019/5261830
11. Neculai A. New Accelerated Conjugate Gradient Algorithms as a Modification of Dai-Yuan's Computational Scheme for Unconstrained Optimization. Journal of Computational and Applied Mathematics, 2010, vol. 234, no. 12, pp. 3397-3410. DOI: 10.1016/j.cam.2010.05.002
12. Eman H., Rana A.Z., Abbas A.Y. New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization. Journal of Mathematics, 2020, vol. 2020, article ID: 3615208, 10 p. DOI: 10.1155/2020/3615208
13. Gaohang Yu, Lutai Guan, Wufan Chen. Spectral Conjugate Gradient Methods with Sufficient Descent Property for Large-Scale Unconstrained Optimization. Optimization Methods and Software, 2008, vol. 23, no. 2, pp. 275-293. DOI: 10.1080/10556780701661344
14. Ibrahim Sulaiman Mohammed, Yakubu Usman Abbas, Mamat Mustafa. Application of Spectral Conjugate Gradient Methods for Solving Unconstrained Optimization Problems. An International Journal of Optimization and Control: Theories and Applications, 2020, vol. 10, no. 2, pp. 198-205.
15. Wang Li, Cao Mingyuan, Xing Funa, Yang Yueting. The New Spectral Conjugate Gradient Method for Large-Scale Unconstrained Optimisation. Journal of Inequalities and Applications, 2020, vol. 2020, no. 1, pp. 1-11. DOI: 10.1186/s13660-020-02375-z
16. Jian Jinbao, Yang Lin, Jiang Xianzhen, Liu Pengjie, Liu Meixing. A Spectral Conjugate Gradient Method with Descent Property. Mathematics, 2020, vol. 8, no. 2, article ID: 280, 13 p. DOI: 10.3390/math8020280
17. Danhausa A.A., Odekunle R.M., Onanaye A.S. A Modified Spectral Conjugate Gradient Method for Solving Unconstrained Minimization Problems. Journal of the Nigerian Mathematical Society, 2020, vol. 39, no. 2, pp. 223-237.
18. Al-Arbo A., Rana Al-Kawaz. A Fast Spectral Conjugate Gradient Method for Solving Nonlinear Optimization Problems. Indonesian Journal of Electrical Engineering and Computer Science, 2021, vol. 21, no. 1, pp. 429-439. DOI: 10.11591/ijeecs.v21.i1.pp429-439
19. Hassan Basim, Jabbar Hawraz. A New Spectral on the Gradient Methods for Unconstrained Optimization Minimization Problem. Journal of Zankoy Sulaimani, 2020, vol. 22, no. 2, pp. 217-224. DOI: 10.17656/jzs.10822
20. Liu J.K., Feng Y.M., Zou L.M. A Spectral Conjugate Gradient Method for Solving Large-Scale Unconstrained Optimization. Computers and Mathematics with Applications, 2019, vol. 77, no. 3, pp. 731-739.
21. Al-Bayati A.Y. A New Family of Self-Scaling Variable Metric Algorithms for Unconstrained Optimization. Journal of Education and Science, 1991, vol. 12, pp. 25-54.
22. Neculai A. An Unconstrained Optimization Test Functions Collection. Advanced Modeling and Optimization, 2008, vol. 10, no. 1, pp. 147-161.
23. Neculai A. New Accelerated Conjugate Gradient Algorithms for Unconstrained Optimization. Technical Report, 2008, vol. 34, pp. 319-330.
24. Dolan E.D., More J.J. Benchmarking Optimization Software with Performance Profiles. Mathematical Programming, 2002, vol. 91, no. 2, pp. 201-213. DOI: 10.1007/s101070100263