具有能级稳定过程的MQHOA优化算法

doi:10.11959/j.issn.1000-436x.2016136

Abstract

Abstract:

An improved multi-scale quantum harmonic oscillator algorithm (MQHOA) with energy level stabilizing process was proposed analogizing to quantum harmonic oscillator's wave function. Inspired by quantum model, the op-timization problem was transformed to finding ground state wave function of bound state. Harmonic oscillator potential well was used to approach objective function under the condition of Taylor approximation. Energy level stabilization, en-ergy level reduction, scale reduction were the basic iterative convergence processes of MQHOA, coinciding with its physical model. Only one subjective control parameter was needed in MQHOA whose wave function and zero-point en-ergy were defined with reference to quantum model. Experimental results show that MQHOA's performance is superior to several other common optimization algorithms. For high dimensional testing functions including Ackley、Griewank、Sphere、Sum Squares、Zakharov, etc, the global optimums can be obtained precisely with 100% probability.

Key words: optimization algorithm, function optimization, MQHOA, wave function, ground state

Peng WANG,Yan HUANG. MQHOA algorithm with energy level stabilizing process[J]. Journal on Communications, 2016, 37(7): 79-86.

Figures/Tables 8

MQHOA算法	量子理论
目标函数 f(x)	势阱V(x)
求 f(x)的全局最优解	求V(x)的基态波函数
f (x)为复杂目标函数	采用 f(x)泰勒展开的二次近似 $V (x) = \frac{1}{2} f ″ (x_{0}) {(x - x_{0})}^{2}$ 逼近
基本采样函数为正态分布	谐振子波函数特性为正态分布
局部最优	高能态下的亚稳态
全局最优	基态
If f( $f (x_{i}^{'}) < f (x_{i})$ )＜f(x_i)	高能级亚稳态稳定过程
then x_i= $x_{i}^{'}$
x^worst=x^mean	由高能态向低能态跃迁
多尺度过程	量子退火过程
当前解与最优解的差值	零点能

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函数编号	函数名
f₁	Ackley
f₂	Dixon ＆ Price
f₃	Griewank
f₄	Levy
f₅	Sphere
f₆	Sum Squares
f₇	Zakharov

目标函数		10维			30维
目标函数	均值	最优值	函数进化次数	均值	最优值	函数进化次数
f₁	7.508 452×10^-6	3.333 165×10^-6	8.878 433×10³	1.105 000×10^-5	7.923 803×10^-6	2.511 313×10⁴
f₂	6.666 667×10^-1	6.666 667×10^-1	7.111 920×10⁴	6.666 667×10^-1	6.666 667×10^-1	4.412 675×10⁵
f₃	3.399 435×10^-12	9.113 821×10^-13	1.569 017×10⁴	2.145 187×10^-11	1.222 245×10^-11	6.248 157×10⁴
f₄	1.435 393×10^-11	2.840 595×10^-12	4.536 853×10⁴	3.885 483×10^-10	8.875 998×10^-11	1.255 304×10⁵
f₅	3.600 803×10^-11	5.812 973×10^-12	8.688 300×10³	2.290 778×10^-10	1.265 690×10^-10	2.427 407×10⁴
f₆	1.534 184×10^-10	4.719 128×10^-11	2.862 647×10⁴	2.294 445×10^-8	4.389 170×10^-9	2.035 047×10⁵
f₇	9.381 916×10^-11	2.710 694×10^-11	2.055 097×10⁴	1.904 055×10^-9	7.740 381×10^-10	1.298 756×10⁵

目标函数	定义域	ABC	JDE	CLPSO	MQHOA
f₁	[-32.768,32.768]	4.65×10^-1	8.87×10^-1	4.42	3.35
f₃	[-600,600]	1.12×10^-1	7.57×10^-2	1.78	2.87×10^-3
f₅	[-100,100]	9.44×10^-2	1.38×10^-8	8.42×10¹	1.42×10^-9
f₆	[-100,100]	3.08	1.79×10^-7	3.88×10³	3.32×10^-6
f₇	[-10,10]	2.53×10³	2.81×10²	1.97×10³	4.85×10^-8
注：D=100,ABC、JDE、CLPSO算法实验数据引自文献[17]。函数的优化性能是优化算法的一个重要评价指标。

目标函数	D	均值	最优值	标准差	成功率	函数进化次数
f₁	500	4.495 454×10^-5	4.169 445×10^-5	1.852 217×10^-6	100%	3.581 637×10⁵
f₃	500	1.819 410×10^-9	1.646 367×10^-9	1.812 600×10^-10	100%	1.179 493×10⁶
f₅	500	6.201 470×10^-8	5.126 464×10^-8	6.219 188×10^-9	100%	3.404 131×10⁵
f₆	200	9.941 020×10^-5	5.342 778×10^-5	2.095 928×10^-5	100%	3.294 248×10⁶
f₇	200	2.691 814×10^-6	1.706 075×10^-6	7.895 281×10^-7	100%	5.942 116×10⁶

MQHOA algorithm with energy level stabilizing process

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