一个新的填充函数及其在数据拟合问题中的应用

doi:10.15960/j.cnki.issn.1007-6093.2021.01.007

Abstract

Abstract:

The filled function method is one of the effective methods to solve the global optimization problem. In this paper, a new continuous and differentiable nonparameter filled function is proposed for the unconstrained optimization problem. The related properties of the filled function are proved and the corresponding algorithm is designed. By comparing with the numerical experimental results in previous literature, it is shown that the proposed filled function algorithm is effective and feasible. Then, the proposed filled function method is used to solve the data fitting example of cutting temperature experimental data, compared with the existing least squares method and genetic algorithm, the fitting effect is better.

Key words: global optimization, filled function, non-parameter, data fitting

CLC Number:

O221.2

Jiali CHEN, Ying ZHANG, Shenggang WANG, Xiaoying XIE. A new filled function and its application in data fitting problems[J]. Operations Research Transactions, 2021, 25(1): 81-88.

Figures/Tables 6

References 11

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	$x_{1}$	$k$	$x^*$	$f^*$
本文	$(-1, 2)$	1	$(1, 1)$	$0.0$
本文	$(-3, -3)$	1	$(0.999\; 999\; 99, 0.999\; 999\; 97)$	$1.626\; 4\times10^{-16}$
文献[8]	$(-2, 2)$	$2$	$(1, 1)$	$1.23\times10^{-10}$
文献[9]	$(-3, 3)$	$\setminus$	$(1, 1)$	$0$

	$x_{1}$	$k$	$x^*$	$f^*$
本文	$(-1, 3)$	2	$(0, 0)$	$0$
本文	$(-3, -2)$	1	$(-2.000\; 0, 9.097\; 6\times10^{-14})$	$8.276\; 6\times10^{-27}$
文献[6]	$(2, 2)$	$1$	$(-2, 0)$	$7.785\; 3\times10^{-13}$
文献[9]	$(-3, 0)$	$\setminus$	$(9.117\; 3\times10^{-9}, 6.731\; 8\times10^{-8})$	$4.864\; 3\times10^{-15}$

	$x_{1}$	$k$	$x^*$	$f^*$
本文	$(2, 3)$	$5$	$(5.482\; 864\; 21, 4.858\; 056\; 88)$	$-186.730\; 9$
	$(-2, -1)$	$2$	$(-0.800\; 321\; 1 , -1.425\; 128\; 43)$	$-186.730\; 9$
	$(-3, -4)$	$2$	$(-1.425\; 128\; 43, -7.083\; 506\; 41)$	$-186.730\; 9$
	$(-5, 4)$	$2$	$(-7.083\; 506\; 41, 4.858\; 056\; 88)$	$-186.730\; 9$
	$(-6, -7)$	$1$	$(-7.708\; 313\; 74, -7.083\; 506\; 41)$	$-186.730\; 9$
文献[7]	$(1, 1)$	$6$	$(-1.425\; 2, -0.800\; 3)$	$-186.730\; 9$

	$c$	$x_{1}$	$k$	$x^*$	$f^*$
本文	c = 0.2	$(1, 2)$	2	$( 1.590\; 8, -0.270\; 2)$	$3.325\; 3\times 10^{15}$
	$c = 0.5$	$(0, 0)$	2	$(1.0, -4.47\times 10^{10})$	$5.132\; 5\times 10^{17}$
	$c = 0.05$	$(1, 3)$	1	$(1.851\; 3, -0.402\; 0)$	$6.460\; 1\times 10^{18}$
文献[7]	$c = 0.2$	$(3, 3)$	$3$	$(1.878\; 5, -0.345\; 8)$	$6.962\; 6\times 10^{9}$
	$c = 0.5$	$(0, 0)$	$3$	$(0.102\; 6, 0.300\; 5)$	$1.260\; 9\times 10^{8}$
	$c = 0.05$	$(6, -2)$	$5$	$(1.851\; 3, -0.402\; 1)$	$4.875\; 1\times 10^{9}$

序号$i$	$\gamma/$mm	$\beta/{\rm (mm\cdot r^{-1})}$	$v/{\rm (m\cdot min^{-1})}$	切削温度T$/ ^\circ {\rm C}$
1	1	0.1	60	740.24
2	1.41	0.2	60	793.80
3	2	0.4	60	854.80
4	1.41	0.1	94.8	819.42
5	2	0.2	94.8	870.00
6	1	0.4	94.8	904.80
7	2	0.1	150	879.60
8	1	0.2	150	911.10
9	1.41	0.4	150	974.10

A new filled function and its application in data fitting problems

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求解方法	最优拟合参数$\theta^*$	剩余标准差$S$
填充函数法	$(470.4, 0.032, 0.082\; 9, 0.159)$	$S = 5.131$
最小二乘法	$(467, 0.033, 0.083, 0.16)$	$S = 5.699$
遗传算法	$(471, 0.030\; 1, 0.083, 0.159)$	$S = 5.20$

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