智能优化算法的量子理论纲要

王鹏; 辛罡

doi:10.16383/j.aas.c190761

智能优化算法的量子理论纲要

doi: 10.16383/j.aas.c190761

王鹏^1,,
辛罡^{2, 3,}

1.
西南民族大学计算机科学与技术学院成都 610225
2.
中国科学院成都计算机应用研究所成都 610041
3.
中国科学院大学北京 100049

基金项目: 西南民族大学中央高校基本科研业务费专项资金项目(2020NYB18)资助

详细信息

作者简介:
王鹏：西南民族大学计算机科学与技术学院教授. 2001年获四川大学核技术及应用专业硕士学位. 2004年获中国科学院成都计算机应用研究所计算机软件与理论专业博士学位. 主要研究方向为量子理论, 量子启发式算法, 计算智能与高性能计算. 本文通信作者. E-mail: qhoalab@163.com

辛罡：中国科学院成都计算机应用研究所博士研究生. 2011年获德累斯顿工业大学微电子与固体电子专业硕士学位. 主要研究方向为量子启发式算法, 高性能计算. E-mail: xin_gang@vip.126.com

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出版历程
- 收稿日期: 2019-11-03
- 录用日期: 2020-02-07
- 网络出版日期: 2023-10-19
- 刊出日期: 2023-11-22

Quantum Theory of Intelligent Optimization Algorithms

WANG Peng^1
,,
XIN Gang^{2, 3
,}

1.
School of Computer Science and Technology, Southwest Minzu University, Chengdu 610225
2.
Chengdu Institute of Computer Application, Chinese Academy of Sciences, Chengdu 610041
3.
University of Chinese Academy of Sciences, Beijing 100049

Funds: Supported by Fundamental Research Funds for the Central Universities, Southwest Minzu University (2020NYB18)

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Author Bio:
WANG Peng　Professor at the School of Computer Science and Technology, Southwest Minzu University. He received his master degree in nuclear technology and applications from Sichuan University in 2001 and Ph.D. degree in computer software and theory from Chengdu Institute of Computer Application, Chinese Academy of Sciences in 2004. His research interest covers quantum mechanics, quantum inspired algorithm, computational intelligence, and high performance computing. Corresponding author of this paper

XIN Gang　Ph.D. candidate at Chengdu Institute of Computer Application, Chinese Academy of Sciences. He received his master degree in microelectronics and solid state electronics from Dresden University of Technology, Germany in 2011. His research interest covers quantum inspired algorithm and high performance computing

摘要

摘要: 针对一些智能优化算法缺乏完备数学物理理论基础的现状, 利用优化问题和量子物理在概率意义上的相似性, 建立优化问题的薛定谔方程, 将优化问题转化为以目标函数为约束条件的基态波函数问题, 同时利用波函数定义了算法的能量、隧道效应和熵, 实现了以波函数为中心的优化问题量子模型. 这一纲要利用了量子物理完备的理论框架, 建立起了优化问题与量子理论广泛的内在联系. 从量子物理的角度回答了优化问题解的概率描述, 邻域采样函数的选择, 算法演化的过程设计, 多尺度过程的必要性等问题. 智能优化算法的量子理论纲要可以作为研究与构造算法的理论工具, 其有效性已得到初步验证.
- 优化问题 /
- 优化算法 /
- 量子理论 /
- 波函数 /
- 基态
Abstract: Many metaheuristics, which are based on the metaphor of natural phenomena, are lacking the support of a complete mathematical or physical theory. In this paper, the Schrödinger equation of optimization problems is proposed based on the basic probabilistic similarities between an optimization problem and a quantum system. It transforms the optimization problem into a quantum physics problem with the objective function as the constrained conditions. Meanwhile, the algorithm energy, quantum tunnel effect, and entropy are defined based on the probabilistic interpretation of the wave function. This wave function based quantum physical model establishes a comprehensive internal relationship between optimization problems and quantum theory, and theoretically answers the following questions: probabilistic description of solution of optimization problems, selection of neighborhood function, evolutionary process of optimization, necessity of multi-scale process, etc. Those quantum theories of intelligent optimization algorithm can be used as a theoretical means for the analysis of optimization systems. Its effectiveness has been preliminarily verified.
- Optimization problem /
- optimization algorithm /
- quantum theory /
- wave function /
- ground state

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图 1 量子隧道效应示意图

Fig. 1 Schematic diagram of quantum tunneling effect

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表 1 PVADE, SPSO2011, QPSO, LoTFWA和MPSS-MQHOA在30维CEC2013测试集下平均误差和标准差的对比实验, 所有算法的终止条件为MaxFES = 300000, 所有实验独立重复51次

Table 1 Comparison of mean errors and standard deviations of PVADE, SPSO2011, QPSO, LoTFWA, MPSS-MQHOA, under 30D CEC2013 benchmark functions. The stopping condition for all schemes is set at MaxFES = 300000. The experiments are repeated 51 times individually

	PVADE		SPSO2011		QPSO		LoTFWA		MPSS-MQHOA
F.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.
1	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00	0.00E + 00
2	4.24E + 04	2.74E + 04	1.06E + 05	4.90E + 04	1.90E + 07	7.65E + 06	2.13E + 06	7.21E + 05	8.96E + 05	2.14E + 05
3	1.39E + 06	4.58E + 06	5.64E + 07	8.68E + 07	3.98E + 08	8.77E + 08	3.94E + 07	7.23E + 07	3.01E + 06	3.21E + 06
4	1.11E + 02	9.05E + 01	1.97E + 03	7.77E + 02	1.20E + 04	2.77E + 03	7.03E + 03	1.97E + 03	1.69E + 04	3.56E + 03
5	1.12E − 04	3.35E − 04	4.72E − 04	3.75E − 05	1.77E − 05	1.51E − 05	1.60E − 02	1.41E − 02	2.40E − 03	3.61E − 04
AR. 1 ~ 5	1.20	1.60	2.40	2.20	3.20	3.20	3.20	3.20	3.00	2.80
6	4.73E + 01	2.49E + 01	1.46E + 01	1.86E + 01	3.87E + 01	1.95E + 01	1.94E + 01	1.25E + 01	2.42E + 01	2.18E + 01
7	7.33E + 00	5.53E + 00	4.32E + 01	1.44E + 01	4.79E + 01	1.57E + 01	5.40E + 01	1.23E + 01	2.63E + 01	1.04E + 01
8	2.09E + 01	4.64E − 02	2.09E + 01	6.31E − 02	2.10E + 01	4.03E − 02	2.09E + 01	6.20E − 02	2.09E + 01	6.50E − 02
9	1.05E + 01	2.37E + 00	2.37E + 01	4.41E + 00	2.55E + 01	8.37E + 00	1.55E + 01	2.03E + 00	1.59E + 01	1.78E + 00
10	5.36E − 01	6.07E − 01	2.47E − 01	1.32E − 01	2.86E + 00	1.61E + 00	5.60E − 02	2.89E − 02	4.48E − 02	3.76E − 02
11	6.12E + 01	1.07E + 01	6.96E + 01	2.36E + 01	1.66E + 02	1.97E + 01	6.59E + 01	1.47E + 01	3.41E + 01	9.77E + 00
12	1.01E + 02	1.53E + 01	5.65E + 01	1.52E + 01	2.04E + 02	1.49E + 01	7.71E + 01	1.64E + 01	3.36E + 01	8.56E + 00
13	1.22E + 02	1.59E + 01	1.19E + 02	2.76E + 01	2.03E + 02	1.21E + 01	1.45E + 02	2.62E + 01	7.36E + 01	1.69E + 01
14	3.20E + 03	3.77E + 02	4.34E + 03	7.89E + 02	6.20E + 03	5.36E + 02	2.69E + 03	3.13E + 02	2.57E + 03	3.32E + 02
15	5.29E + 03	3.90E + 02	4.30E + 03	5.99E + 02	7.25E + 03	3.07E + 02	2.81E + 03	3.89E + 02	2.50E + 03	3.16E + 02
16	2.32E + 00	3.06E − 01	1.74E + 00	3.06E − 01	2.50E + 00	2.78E − 01	6.74E − 02	2.44E − 02	1.36E − 01	3.84E − 02
17	9.39E + 01	1.24E + 01	1.32E + 02	3.20E + 01	2.36E + 02	1.61E + 01	6.60E + 01	1.05E + 01	5.56E + 01	5.96E + 00
18	1.67E + 02	1.13E + 01	1.20E + 02	1.65E + 01	2.42E + 02	1.42E + 01	6.82E + 01	1.01E + 01	5.55E + 01	6.94E + 00
19	6.48E + 00	2.62E + 00	6.95E + 00	3.36E + 00	1.76E + 01	1.74E + 00	3.48E + 00	7.68E − 01	2.24E + 00	5.63E − 01
20	1.34E + 01	1.93E + 00	1.30E + 01	1.94E + 00	1.24E + 01	4.08E − 01	1.32E + 01	9.99E − 01	1.36E + 01	1.04E + 00
21	3.26E + 02	8.62E + 01	3.35E + 02	6.63E + 01	2.86E + 02	7.88E + 01	1.98E + 02	1.41E + 01	3.02E + 02	3.46E + 01
22	2.53E + 03	5.87E + 02	3.97E + 03	8.07E + 02	6.37E + 03	4.79E + 02	3.08E + 03	3.59E + 02	3.07E + 03	4.17E + 02
23	5.21E + 03	4.82E + 02	4.32E + 03	7.14E + 02	7.25E + 03	3.01E + 02	3.30E + 03	4.26E + 02	3.32E + 03	5.49E + 02
24	2.24E + 02	1.23E + 01	2.46E + 02	9.15E + 00	2.46E + 02	9.01E + 00	2.43E + 02	9.81E + 00	2.19E + 02	4.58E + 00
25	2.49E + 02	2.12E + 01	2.77E + 02	9.40E + 00	2.59E + 02	8.05E + 00	2.80E + 02	8.07E + 00	2.66E + 02	1.12E + 01
26	2.49E + 02	6.12E + 01	2.69E + 02	7.02E + 01	2.96E + 02	6.60E + 01	2.00E + 02	3.48E − 02	2.00E + 02	3.55E − 02
27	5.21E + 02	8.32E + 01	7.31E + 02	1.03E + 02	7.52E + 02	1.03E + 02	7.21E + 02	1.15E + 02	5.38E + 02	4.34E + 01
28	2.73E + 02	1.80E + 02	3.22E + 02	1.57E + 02	3.93E + 02	3.18E + 02	2.84E + 02	5.47E + 01	3.00E + 02	8.96E − 09
AR. 6 ~ 28	2.87	3.43	3.35	4.13	4.48	3.04	2.43	2.30	1.87	2.09
AR. 1 ~ 28	2.57	3.11	3.18	3.79	4.25	3.07	2.57	2.46	2.07	2.21

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表 2 优化算法与量子理论的对应关系

Table 2 The relationship between optimization algorithm and quantum theory

优化算法	量子理论	说明
优化问题的解	波函数$ \psi(x)$	优化问题的概率描述
优化问题	薛定鄂方程的约束条件	通过势阱等效将优化问题引入量子系统
优化系统的动态模型	含时薛定鄂方程	含时薛定鄂方程描述了随机搜索过程
多尺度过程	不确定性关系	通过算符的概念证明了多尺度过程的必要性
群体策略	波函数的叠加性	通过波函数的叠加性解释了算法种群的必要性
跳出局部最优	量子隧道效应	可以通过波函数来计算隧道效应概率
解的确定性	波函数熵	通过波函数定义优化算法的熵

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