Riemannian流形中DE算法算子最优特征量的量子渐进估计

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 31-43.

Riemannian流形中DE算法算子最优特征量的量子渐进估计

王凯光¹(),高岳林^2,*

¹ 北方民族大学数学与信息科学学院银川 750021
² 北方民族大学宁夏智能信息与大数据处理重点实验室银川 750021

收稿日期:2019-04-01 出版日期:2020-01-26 发布日期:2020-04-08
通讯作者: 高岳林 E-mail:wkg13759842420@foxmail.com
作者简介:王凯光, E-mail:wkg13759842420@foxmail.com
基金资助:
国家自然科学基金(61561001);北方民族大学重大科研专项资助项目(ZDZX201901);北方民族大学研究生创新项目(YCX19120);宁夏高等教育一流学科建设资助项目(NXYLXK2017B09)

Quantum Asymptotic Estimation of the Optimal Eigenvalues of DE Operators in Riemannian Manifolds

Kaiguang Wang¹(),Yuelin Gao^2,*

¹ School of Mathematics and Information Science, North Minzu University, Yinchuan 750021
² Ningxia Key Laboratory of Intelligent Information and Big Data Processing, North Minzu University, Yinchuan 750021

Received:2019-04-01 Online:2020-01-26 Published:2020-04-08
Contact: Yuelin Gao E-mail:wkg13759842420@foxmail.com
Supported by:
国家自然科学基金(61561001);北方民族大学重大科研专项资助项目(ZDZX201901);北方民族大学研究生创新项目(YCX19120);宁夏高等教育一流学科建设资助项目(NXYLXK2017B09)

摘要/Abstract

摘要：

该文主要分析和探讨了差分进化算法（Differential Eveolutionary Algorithm，DE）在Riemannian流形中的几何关系，对P_-ε条件下Riemannian流形中的种群个体进行了收敛性分析，得到了迭代个体收敛精度与收敛速度的量子不确定渐进估计，如下式

$\Delta_{v}^{2}\cdot \Delta_{x_{\beta}^{\varepsilon}}^{2}\geq\Big(\frac{\sqrt{(\lambda_{\varepsilon})_{1}}+\cdots +\sqrt{(\lambda_{\varepsilon})_{n}}}{2}\Big)^{2},$

其中，Δ_v²为种群个体的速度分辨率，Δ_{x_β^ε}²为种群个体带有误差的位置分辨率，（λ_ε）_i，i=1，2，…，n.从本质上说明了Riemannian流形中迭代个体的局部特征量是不能从收敛精度和收敛速度同时达到算法高效.

关键词: DE算法, Riemannian流形, 收敛精度, 收敛速度, 量子不确定渐进估计

Abstract:

In this paper, the geometric relations of differential evolution algorithm in Riemannian manifolds are analyzed and discussed. The convergence of populations in Riemannian manifolds with P-ε is analyzed. A quantum uncertain asymptotic estimation of the convergence accuracy and convergence speed of the iterative individual is obtained as follows

$\Delta_{v}^{2}\cdot \Delta_{x_{\beta}^{\varepsilon}}^{2}\geq\Big(\frac{\sqrt{(\lambda_{\varepsilon})_{1}}+\cdots +\sqrt{(\lambda_{\varepsilon})_{n}}}{2}\Big)^{2},$

where, Δ_v² is speed resolution of individual populations, Δ_{x_β^ε}² is position resolution with error ε of individual populations, (λ_ε)_i, i=1, 2, …, n. The theorem expression essentially shows that the local eigenvalues of iterated individuals in Riemann manifolds can not achieve high convergent accuracy and convergent speed at the same time.

Key words: DE algorithm, Riemannian manifolds, Convergent accuracy, Convergent speed, Quantum uncertain asymptotic estimation

中图分类号:

O192

王凯光,高岳林. Riemannian流形中DE算法算子最优特征量的量子渐进估计[J]. 数学物理学报, 2020, 40(1): 31-43.

Kaiguang Wang,Yuelin Gao. Quantum Asymptotic Estimation of the Optimal Eigenvalues of DE Operators in Riemannian Manifolds[J]. Acta mathematica scientia,Series A, 2020, 40(1): 31-43.

参考文献 19

1	Storn R , Price K . Differential evolution -a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 1997, 11 (4): 341- 359
2	Yuan X , Cao B , Yang B , et al. A modified differential evolution for constrained optimization. Information Computing and Automation, 2008, 3, 19- 22
3	Qing A. Advances in Differential Evolution. Germany: Springer Berlin Heidelberg, 2008
4	Rönkkönen J , Kukkonen S , Price K V . Real-parameter optimization with differential evolution. IEEE Congress on Evolutionary Computation, 2005, 1 (1): 506- 513
5	Price K, Storn R M, Lampinen J A. Differential Evolution: A Practical Approach to Global Optimization. Germany: Springer Science and Business Media, 2006
6	王凯光, 高岳林. 十进制整数编码的DE算法模式集定理研究. 应用数学, 2019, 32 (02): 443- 451
	Wang K G , Gao Y L . The schema sets theorem of DE algorithm for decimal integer coding. Mathematica Applicata, 2019, 32 (02): 443- 451
7	Qin A K , Huang V L , Suganthan P N . Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE transactions on Evolutionary Computation, 2008, 13 (2): 398- 417
8	Das S , Suganthan P N . Differential evolution:a survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation, 2011, 15 (1): 4- 31
9	Storn R . System design by constraint adaptation and differential evolution. IEEE Transactions on Evolutionary Computation, 1999, 3 (1): 22- 34
10	Thosen R. Flexible Ligand Docking Using Differential Evolution[C]//Proceedings of the 2003 IEEE Congress on Evolutionary Computation. Canberra, Australia, 2003, 4: 2354-2361
11	Yang M , Li C H , Cai Z H , et al. Differential evolution with auto-enhanced population diversity. IEEE Transactions on Cybernetics, 2015, 45 (2): 302- 315
12	Lin T , Zha H . Riemannian manifold learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2008, 30 (5): 796- 809
13	汪容. 数学物理中的微分几何与拓扑学. 杭州: 浙江大学出版社, 2010
	Wang R . Differential Geometry and Topology in Mathematical Physic. Hangzhou: Zhejiang University Press, 2010
14	Busch P , Heinonen T , Lahti P . Heisenberg's uncertainty principle. Physics Reports, 2007, 452 (6): 155- 176
15	Moshinsky M , Quesne C . Linear canonical transformations and their unitary representations. Journal of Mathematical Physics, 1971, 12 (8): 1772- 1780
16	Zhao J , Tao R , Wang Y . On signal moments and uncertainty relations associated with linear canonical transform. Signal Processing, 2010, 90 (9): 2686- 2689
17	庞学诚, 等. 数学分析(第四版·上). 北京: 高等教育出版社, 2001
	Pang X C , et al. Mathematical Analysis (Fourth Edition Part I). Beijing: Higher Education Press, 2001
18	Bracewell R N , Bracewell R N . The Fourier Transform and Its Applications. New York: McGraw-Hill, 1986
19	Wang K G , Gao Y L . Topology structure implied in β-Hilbert space, Heisenberg uncertainty quantum characteristics and numerical simulation of the DE algorithm. Mathematics, 2019, 7 (4): 330- 349

[1]	姚庆六. 一个非线性分数微分方程奇异解的存在性与逐次迭代方法[J]. 数学物理学报, 2016, 36(2): 287-296.
[2]	陶宝. 负极值指标估计量的渐近性质[J]. 数学物理学报, 2014, 34(3): 611-618.
[3]	凌能祥, 丁洁. 相依函数型数据条件密度估计的渐近性质[J]. 数学物理学报, 2012, 32(3): 547-556.
[4]	涂天亮, 莫炯. 插值逼近具边界数据的调和函数[J]. 数学物理学报, 2010, 30(2): 397-404.
[5]	沈陆明, 张继宏, 镇志勇. 形式Laurent级数域上交错Oppenheim展开的研究[J]. 数学物理学报, 2010, 30(1): 207-216.
[6]	邢国东, 杨善朝. 负相协随机变量的指数不等式[J]. 数学物理学报, 2009, 29(6): 1679-1688.
[7]	席福宝. 带马氏切换的马氏过程的指数收敛速度[J]. 数学物理学报, 2009, 29(4): 1051-1057.
[8]	尹长明, 李永明, 王朋炎. 广义线性模型中极大拟似然估计的强收敛速度[J]. 数学物理学报, 2009, 29(4): 1058-1064.
[9]	唐庆国; 王金德. 系数函数光滑度互不相同的变系数模型的一步估计法[J]. 数学物理学报, 2008, 28(4): 701-710.
[10]	甘师信;陈平炎. NOD序列加权和的强收敛速度[J]. 数学物理学报, 2008, 28(2): 283-290.
[11]	孙燕, 柴根象. 固定设计下回归函数的小波估计[J]. 数学物理学报, 2004, 4(5): 597-606.
[12]	薛留根. 混合误差下回归函数小波估计的一致收敛速度[J]. 数学物理学报, 2002, 22(4): 528-535.
[13]	杨筱菡, 柴根象. 相依样本下线性模型误差分布的相合估计[J]. 数学物理学报, 2002, 22(4): 536-547.
[14]	薛留根, 胡玉萍. 条件t-分位数核估计的逼近速度[J]. 数学物理学报, 2001, 21(2): 215-224.
[15]	朱仲义, 韦博成. 条件泛函及其导数非参数估计的强一致收敛速度[J]. 数学物理学报, 1998, 18(4): 394-401.