使用遗传算法编码和最小化健身函数

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This example shows how to create and minimize a fitness function for the genetic algorithm solvergausing three techniques:

基本的
Including additional parameters
Vectorized for speed

基本的Fitness Function

这basic fitness function is Rosenbrock's function, a common test function for optimizers. The function is a sum of squares:

$f (x) = 100 (x_{1}^{2} - x_{2})^{2} + (1 - x_{1})^{2} 。$

该函数在该点的最小值为零[1,1]。Because the Rosenbrock function is quite steep, plot the logarithm of one plus the function.

fsurf(@(x,y)log(1 + 100*(x.^2 - y).^2 + (1 - x).^2),[0,2]) title('log(1 + 100*(x(1)^2 - x(2))^2 + (1 - x(1))^2)') view(-13,78) hold在h1 = plot3(1,1,0.1,'r*',“标记”,12); legend(h1,'Minimum','地点','best'）；抓住离开

Fitness Function Code

这simple_fitnessfunction file implements Rosenbrock's function.

typesimple_fitness

function y = simple_fitness(x) %SIMPLE_FITNESS fitness function for GA % Copyright 2004 The MathWorks, Inc. y = 100 * (x(1)^2 - x(2)) ^2 + (1 - x(1))^2;

健身功能必须接受一个输入xwherex是一个行矢量，其元素与问题中的变量数量一样多。健身函数计算函数的值，并在其一个返回参数中返回标量值y。

Minimize Using`ga`

使用ga, pass a function handle to the fitness function as well as the number of variables in the problem. To havegaexamine the relevant region, include bounds-3 <= x（i）<= 3。Pass the bounds as the fifth and sixth arguments after数字变量。Forgasyntax details, seega。

ga是随机算法。对于可重复性，请设置随机数流。

RNGdefault％可再现性fitnessfunction = @simple_fitness;numberOfvariables = 2;lb = [-3，-3];ub = [3,3];[x，fval] = ga（fitnessfunction，numberOfvariables，[]，[]，[]，[]，[]，lb，ub）

Optimization terminated: maximum number of generations exceeded.

x =1×21.5083 2.2781

fval = 0.2594

这x求解器返回是最终人口中的最佳点ga。这fvalis the value of the functionsimple_fitnessevaluated at the pointx。ga没有找到特别好的解决方案。有关改进解决方案的方法，请参阅遗传算法选择的影响。

Fitness Function with Additional Parameters

Sometimes your fitness function has extra parameters that act as constants during the optimization. For example, a generalized Rosenbrock's function can have extra parameters representing the constants 100 and 1:

$f (x, a, b) = a (x_{1}^{2} - x_{2})^{2} + (b - x_{1})^{2} 。$

a和b是在优化过程中充当常数的适应性函数的参数（它们不是最小化的一部分）。这parameterized_fitness.m文件实现此参数化健身功能。

typeparameterized_fitness

函数y = parameterized_fitness（x，p1，p2）％parameterized_fitness for ga％版权所有2004 Mathworks，Inc。y = p1 *（x（1） ^2 -x（2 -x（2）） ^2 +（p2 -x（p2 -x）1））^2;

Minimize Using Additional Parameters

Use an anonymous function to capture the values of the additional arguments, namely, the constantsa和b。Create a function handleFitnessFunctionto an anonymous function that takes one inputx，和callsparameterized_fitnesswithx,a，和b。这anonymous function contains the values of a and b that exist when the function handle is created.

a = 100; b = 1;% define constant valuesfitnessfunction = @（x）parameterized_fitness（x，a，b）;[x，fval] = ga（fitnessfunction，numberOfvariables，[]，[]，[]，[]，[]，lb，ub）

Optimization terminated: maximum number of generations exceeded.

x =1×21.3198 1.7434

fval = 0.1025

看传递额外的参数。

矢量化适应性函数

为了提高速度，矢量化your fitness function. A vectorized fitness function computes the fitness of a collection of points at once, which generally saves time over evaluating these points individually. To write a vectorized fitness function, have your function accept a matrix, where each matrix row represents one point, and have the fitness function return a column vector of fitness function values.

To change theparameterized_fitnessfunction file to a vectorized form:

Change each variablex（i）tox（：，i），意思是变量的列向量对应于x（i）。
Change each vector multiplication*to。*和each exponentiation^to。^表明操作是元素的。此代码中没有矢量乘法，因此只需更改指数即可。

typevectorized_fitness

函数y = vectorized_fitness（x，p1，p2）％vectorized_fitness fitness for ga％版权所有2004-2010 The MathWorks，Inc。y = p1 *（x（x（：，1）。^2 -x（：，2））。^2 +（p2 -x（：，1））。^2;

此矢量化版本的健身功能采用矩阵xwith an arbitrary number of points, meaning and arbitrary number of rows, and returns a column vectorywith the same number of rows asx。

告诉健身的解决者function is vectorized in the'UseVectorized'选项。

options = optimoptions(@ga,'UseVectorized',true);

将选项作为最后一个参数ga。

VFitnessFunction = @(x) vectorized_fitness(x,100,1); [x,fval] = ga(VFitnessFunction,numberOfVariables,[],[],[],[],lb,ub,[],options)

Optimization terminated: maximum number of generations exceeded.

x =1×21.6219 2.6334

FVAL = 0.3876

速度有什么区别？时间在有或没有矢量化的情况下进行优化。

tic [x，fval] = ga（vfitnessfunction，numberOfvariables，[]，[]，[]，[]，[]，lb，ub，[]，options）;

Optimization terminated: maximum number of generations exceeded.

v = toc;tic [x，fval] = ga（fitnessfunction，numberOfvariables，[]，[]，[]，[]，[]，lb，ub）;

Optimization terminated: maximum number of generations exceeded.

nv = toc;fprintf（'使用矢量化需要％f秒。没有矢量化需要％f秒。\ n'，V，NV）

Using vectorization took 0.153337 seconds. No vectorization took 0.212880 seconds.

在这种情况下，矢量化的改进不是很好，因为计算健身函数所需的时间很少。但是，对于更多耗时的健身功能，矢量化可能会有所帮助。看Vectorize the Fitness Function。