# 一类新的无参数填充函数及其在最小二乘法中的应用A New Type of Non-Parameter Filled Function and Its Application in Least Squares

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In this paper, a novel non-parameter filled function for solving general constrained global optimi-zation problem is proposed. Then the theoretical properties of the function are argued and corre-sponding algorithm is given in this paper. Numerical experiments using the Python programming language and comparison with previous results show that the proposed filled function algorithm is not only effective but also works better when dealing with certain function. Furthermore, the algorithm is tentatively combined with the treatment of the reaction rate and reactant concentra-tion of the enzymatic reaction of chemical experiments, indicating that the algorithm also has good adaptability in practical cases.

1. 引言

2. 新的无参数填充函数

$\left(P\right)\left\{\begin{array}{l}\mathrm{min}f\left(x\right),\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}x\in {R}^{n}.\end{array}$ (2.1)

$\left\{\begin{array}{l}\mathrm{min}f\left(x\right),\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}x\in X\end{array}$ (2.2)

1) ${x}^{*}$$p\left(x,{x}^{*}\right)$ 的一个严格局部极大点， $f\left(x\right)$ 在点 ${x}^{*}$ 处的盆谷 ${B}^{*}$ 成为 $p\left(x,{x}^{*}\right)$ 的峰的一部分。

2) $p\left(x,{x}^{*}\right)$ 在比 ${B}^{*}$ 高的盆谷里没有平稳点，即 $\nabla p\left(x,{x}^{*}\right)\ne 0$

3) 如果存在比 ${B}^{*}$ 低的盆谷 ${B}_{1}^{*}$ ，则在 ${x}^{\prime }$${x}^{*}$ 的连线上极小化 $p\left(x,{x}^{*}\right)$ 得到极小点 ${x}^{″}\in {B}^{*}$ ，其中， $N\left({x}_{1}^{*},{\delta }_{1}\right)\left({\delta }_{1}>0\right)，{x}^{\prime }\in N\left({x}_{1}^{*},{\delta }_{1}\right)$

1) ${x}^{*}\in {B}^{*}$

2) 对于任意一点 $x\in {B}^{*}$ 使得 $x\ne {x}^{*}$$f\left(x\right)>f\left({x}^{*}\right)$ ，存在一条从 $x$ 到c的下降路径。

$\begin{array}{r}\hfill p\left(x,{x}^{*}\right)=-\mathrm{ln}\left(1+{‖x-{x}^{*}‖}^{2}\right)\left(f\left(x\right)-f\left({x}^{*}\right)\right).\end{array}$ (2.3)

$\begin{array}{c}p\left(x,{x}^{*}\right)=-\mathrm{ln}\left(1+‖x-{x}^{*}‖\right)\left(f\left(x\right)-f\left({x}^{*}\right)\right)\\ <0=p\left({x}^{*},{x}^{*}\right)\end{array}$

$f\left(x\right)=f\left({x}^{*}\right)+{\left(x-{x}^{*}\right)}^{T}\nabla f\left(x\right)+o\left(‖x-{x}^{*}‖\right)$

$f\left(x\right)-f\left({x}^{*}\right)={\left(x-{x}^{*}\right)}^{T}\nabla f\left(x\right)+o\left(‖x-{x}^{*}‖\right)>0$

$\nabla p\left(x,{x}^{*}\right)=-\left[\frac{2\left(x-{x}^{*}\right)}{1+{‖x-{x}^{*}‖}^{2}}\left(f\left(x\right)-f\left({x}^{*}\right)\right)+\mathrm{ln}\left(1+{‖x-{x}^{*}‖}^{2}\right)\nabla f\left(x\right)\right]$

$\begin{array}{c}\nabla p{\left(x,{x}^{*}\right)}^{T}\left(x-{x}^{*}\right)=\left[\frac{2{‖x-{x}^{*}‖}^{2}}{1+{‖x-{x}^{*}‖}^{2}}\left(f\left(x\right)-f\left({x}^{*}\right)\right)+\mathrm{ln}\left(1+{‖x-{x}^{*}‖}^{2}\right)\nabla f{\left(x\right)}^{T}\left(x-{x}^{*}\right)\right]\\ <0\end{array}$

$p\left(x,{x}^{*}\right)=-\mathrm{ln}\left(1+{‖x-{x}^{*}‖}^{2}\right)\left(f\left(x\right)-f\left({x}^{*}\right)\right)<0$

$p\left({x}^{\prime },{x}^{*}\right)=-\mathrm{ln}\left(1+{‖{x}^{\prime }-{x}^{*}‖}^{2}\right)\left(f\left({x}^{\prime }\right)-f\left({x}^{*}\right)\right)>0$

3. 填充函数算法

${x}^{*}+\delta \left(1,0\right),{x}^{*}+\delta \left(0,1\right),{x}^{*}-\delta \left(1,0\right),{x}^{*}-\delta \left(0,1\right)$

4. 数值实验

4.1. 测试函数

1) 6-Hump Back Camel函数 [10]

$\left\{\begin{array}{l}\mathrm{min}f\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}-{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}-3\le {x}_{1},{x}_{2}\le 3\end{array}$

2) Restrign函数 [10]

$\left\{\begin{array}{l}\mathrm{min}f\left(x\right)={x}_{1}^{2}+{x}_{2}^{2}-\mathrm{cos}\left(18{x}_{1}\right)-\mathrm{cos}\left(18{x}_{2}\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}-1\le {x}_{1},{x}_{2}\le 1\end{array}$

3) 文献 [11] 中的一个二维函数(c = 0.05)

$\left\{\begin{array}{l}\mathrm{min}f\left(x\right)={\left[1-2{x}_{2}+c*\mathrm{sin}\left(4\text{π}{x}_{2}\right)-{x}_{1}\right]}^{2}+{\left[{x}_{2}-0.5\mathrm{sin}\left(2\text{π}{x}_{1}\right)\right]}^{2}\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}-10\le {x}_{1},{x}_{2}\le 10\end{array}$

4.2. 数值结果

$k$ ：求解第 $k$ 次局部极小点的迭代步数；

${x}_{k}$ ：满足 ${x}_{k}\in X$ 的第 $k$ 次初始点；

${x}_{k}^{*}$ ：第 $k$ 个局部极小点；

$f\left({x}_{k}^{*}\right)$ ：第 $k$ 个局部极小点的目标函数值；

$\text{iter}$ ：算法迭代步数；

${x}^{*}$ ：算法最终得到的全局极小点；

$f\left({x}^{*}\right)$ ：算法最终得到的全局极小值。

Table 1. Original value x = ( 1 , 2 ) T

Table 2. Original value x = ( 1 , 1 ) T

Table 3. c = 0.05 ，Original value x = ( 6 , − 2 ) T

Table 4. Compared with paper [11]

5. 填充函数在实验数据处理中的应用

5.1. 背景和问题

Table 5. Reaction rate and substrate concentration data in puromycin experiments

Table 6. Average the data after treatment with scorpion toxin

5.2. 分析与假设

$y=f\left(x,\beta \right)={\beta }_{1}\left(1-{e}^{-{\beta }_{2}x}\right)$

$\begin{array}{c}\mathrm{min}f\left(x\right)={\underset{i=1}{\overset{n}{\sum }}\left({y}_{i}-{\stackrel{^}{y}}_{i}\right)}^{2}\\ =\underset{i=1}{\overset{n}{\sum }}{\left({\beta }_{1}\left(1-{e}^{-{\beta }_{2}{x}_{\text{​}}}\right)-{\stackrel{^}{y}}_{i}\right)}^{2}\end{array}$

Figure 1. Fitting curve

6. 结语

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