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一种充分下降的修正PRP三项共轭梯度算法
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Abstract:
共轭梯度算法是求解大规模无约束优化问题最有效的方法之一。文章提出一种新型的修正PRP三项共轭梯度算法，该算法具有不依赖任何线搜索充分下降的特点，搜索方向具有信赖域特征。在较为温和条件下，算法全局收敛。数值实验表明，新算法是有效的，比传统PRP三项共轭梯度算法更具竞争力。
The conjugate gradient algorithm is one of the most effective methods for solving large-scale unconstrained optimization problems. This paper proposes a novel modified Polak-Ribière-Polyak (PRP) three-term conjugate gradient algorithm, which possesses the characteristic of ensuring sufficient descent without relying on any line search conditions. The search direction of this algorithm exhibits trust region properties. Under relatively mild conditions, the algorithm achieves global convergence. Numerical experiments demonstrate that the new algorithm is effective and more competitive compared to the classical PRP three-term conjugate gradient algorithm.
| [1] | 胡午杰, 袁功林. 求解非线性方程组的一种三项共轭梯度法[J]. 广西大学学报(自然科学版), 2019, 44(5): 1485-1490. |
| [2] | Andrei, N. (2008) Hybrid Conjugate Gradient Algorithm for Unconstrained Optimization. Journal of Optimization Theory and Applications, 141, 249-264. https://doi.org/10.1007/s10957-008-9505-0 |
| [3] | Livieris, I.E., Tampakas, V. and Pintelas, P. (2018) A Descent Hybrid Conjugate Gradient Method Based on the Memoryless BFGS Update. Numerical Algorithms, 79, 1169-1185. https://doi.org/10.1007/s11075-018-0479-1 |
| [4] | Djordjević, S.S. (2019) New Hybrid Conjugate Gradient Method as a Convex Combination of Ls and Fr Methods. Acta Mathematica Scientia, 39, 214-228. https://doi.org/10.1007/s10473-019-0117-6 |
| [5] | Jian, J., Chen, Q., Jiang, X., Zeng, Y. and Yin, J. (2016) A New Spectral Conjugate Gradient Method for Large-Scale Unconstrained Optimization. Optimization Methods and Software, 32, 503-515. https://doi.org/10.1080/10556788.2016.1225213 |
| [6] | Faramarzi, P. and Amini, K. (2019) A Modified Spectral Conjugate Gradient Method with Global Convergence. Journal of Optimization Theory and Applications, 182, 667-690. https://doi.org/10.1007/s10957-019-01527-6 |
| [7] | Fatemi, M. (2016) A Scaled Conjugate Gradient Method for Nonlinear Unconstrained Optimization. Optimization Methods and Software, 32, 1095-1112. https://doi.org/10.1080/10556788.2016.1233971 |
| [8] | Beale, E. (1972) A Derivation Conjugate Gradient. Academic Press, 39-43. |
| [9] | Yuan, G., Wei, Z. and Li, G. (2014) A Modified Polak-Ribière-Polyak Conjugate Gradient Algorithm for Nonsmooth Convex Programs. Journal of Computational and Applied Mathematics, 255, 86-96. https://doi.org/10.1016/j.cam.2013.04.032 |
| [10] | Andrei, N. (2013) A Simple Three-Term Conjugate Gradient Algorithm for Unconstrained Optimization. Journal of Computational and Applied Mathematics, 241, 19-29. https://doi.org/10.1016/j.cam.2012.10.002 |
| [11] | Zhang, L., Zhou, W. and Li, D. (2006) A Descent Modified Polak-Ribière-Polyak Conjugate Gradient Method and Its Global Convergence. IMA Journal of Numerical Analysis, 26, 629-640. https://doi.org/10.1093/imanum/drl016 |
| [12] | Yuan, G. and Zhang, M. (2015) A Three-Terms Polak-Ribière-Polyak Conjugate Gradient Algorithm for Large-Scale Nonlinear Equations. Journal of Computational and Applied Mathematics, 286, 186-195. https://doi.org/10.1016/j.cam.2015.03.014 |
| [13] | 王松华, 黎勇. 求解大规模无约束优化问题的一种新的PRP三项共轭梯度法[J]. 广西民族大学学报(自然科学版), 2018, 24(4): 58-66 |
| [14] | Fletcher, R. (1964) Function Minimization by Conjugate Gradients. The Computer Journal, 7, 149-154. https://doi.org/10.1093/comjnl/7.2.149 |
| [15] | Dai, Y.H. and Yuan, Y. (1999) A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property. SIAM Journal on Optimization, 10, 177-182. https://doi.org/10.1137/s1052623497318992 |
| [16] | 袁亚湘, 孙文瑜. 最优化理论与方法[M]. 北京: 科学出版社, 1997: 183-218. |
| [17] | Andrei, N. (2008) An Unconstrained Optimization Test Functions Collection. Environmental Science and Technology, 10, 6552-6558. |
| [18] | Bongartz, I., Conn, A.R., Gould, N. and Toint, P.L. (1995) CUTE: Constrained and Unconstrained Testing Environment. ACM Transactions on Mathematical Software, 21, 123-160. https://doi.org/10.1145/200979.201043 |
| [19] | Dolan, E.D. and Moré, J.J. (2002) Benchmarking Optimization Software with Performance Profiles. Mathematical Programming, 91, 201-213. https://doi.org/10.1007/s101070100263 |
