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MATLAB遗传算法及其实现

遗传算法以一种群体中的所有个体为对象，并利用随机化技术指导对一个被编码的参数空间进行高效搜索。

其中，选择、交叉和变异构成了遗传算法的遗传操作；参数编码、初始群体的设定、适应度函数的设计、遗传操作设计、控制参数设定等5个要素组成了遗传算法的核心内容。

主要步骤如下所示：    

(1)编码：GA在进行搜索之前先将解空间的解数据表示成遗传空间的基因型串结构数据，这些串结构数据的不同组合便构成了不同的点。

(2)初始群体的生成：随机产生N个初始串结构数据，每个串结构数据称为一个个体，N个个体构成了—个群体。

GA以这N个串结构数据作为初始点开始迭代。

(3)适应性值评估检测：适应性函数表明个体或解的优劣性。对于不同的问题，适应性函数的定义方式也不同。

(4)选择：选择的目的是为了从当前群体个选出优良的个体，使它们有机会作为父代为下一代繁殖子孙。

遗传算法通过选择过程体现这一思想，进行选择的原则是适应性强的个体为下一代贡献一个或多个后代的概率大。

选择实现了达尔文的适者生存原则。

(5)交叉：交叉操作是遗传算法中最主要的遗传操作。

通过交叉操作可以得到新一代个体，新个体组合了其父辈个体的特性。交叉体现了信息交换的思想。

(6)变异：变异首先在群体中随机选择一个个体，对于选中的个体以一定的概率随机地改变串结构数据中某个串的值。

同生物界一样，GA中变异发生的概率很低，通常取值在0.001~0.01之间。变异为新个体的产中提供了机会。

壹

遗传算法的MATLAB实现

需要如下主函数：

编码和种群生成

 function [pop] = initializega(num,bounds,evalFN,evalOps,options)

% pop    - the initial, evaluated, random population

% num    - the size of the population, i.e. the number to create

% bounds - the number of permutations in an individual (e.g., number

%          of cities in a tsp

% evalFN - the evaluation fn, usually the name of the .m file for evaluation

% evalOps- any options to be passed to the eval function defaults [ ]

% options- options to the initialize function, ie. [eps, float/binary, prec]

%        where eps is the epsilon value and the second option is 1 for

%     orderOps, prec is the precision of the variables defaults [1e-6 1]

01

交叉

function [c1,c2] = arithXover(p1,p2,bounds,Ops)

% Arith crossover takes two parents P1,P2 and performs an interpolation

% along the line formed by the two parents.

%

% function [c1,c2] = arithXover(p1,p2,bounds,Ops)

% p1      - the first parent ( [solution string function value] )

% p2      - the second parent ( [solution string function value] )

% bounds  - the bounds matrix for the solution space

% Ops     - Options matrix for arith crossover [gen #ArithXovers]

02

选择

normGeomSelect：NormGeomSelect is a ranking selection

function based on the normalized geometric distribution.

(基于正态分布的序列选择函数)

03

变异

function[newPop] = normGeomSelect(oldPop,options)

% NormGeomSelect is a ranking selection function based on

the normalized

% geometric distribution.

%

% function[newPop] = normGeomSelect(oldPop,options)

% newPop  - the new population selected from the oldPop

% oldPop  - the current population

% options - options to normGeomSelect

[gen probability_of_selecting_best]

04

一些辅助函数：

f2b ：Return the binary representation of the float number

fval(将浮点数转化为二进制数)

b2f：Return the float number corresponing to the binary

representation of bval. (将二进制数转化为

浮点数)

nonUnifMutation： Non uniform mutation changes one

of the parameters of the parent based on a non-uniform

probability distribution.  This Gaussian distribution starts wide,

and narrows to a point distribution as the current generation

approaches the maximum generation.

(基于非均一概率分布进行非均一变异)

maxGenTerm：Returns 1, i.e. terminates the GA when the

maximal_generation is reached.

(当迭代次数大于最大迭代次数时，终止遗传算法，返回

为1，否则返回为0。)

roulette：roulette is the traditional selection function with the

probability of surviving equal to the fittness of i / sum of the

fittness of all individuals

贰

应用举例

01

1.计算下列函数的最大值。

f(x)=x+10*sin(5x)+7cos(4x) , x∈[0,9]

方式1  >>gademo

方式2

step 1 编写目标函数gademo1eval1.m

function [sol, val] = gaDemo1Eval(sol,options)

x=sol(1);

val = x + 10*sin(5*x)+7*cos(4*x);

step 2 生成初始种群，大小为10

initPop=initializega(10,[0, 9],'gademo1eval1',[],[1e-6,1]);

step 3  25次遗传迭代

[x, endPop,bpop,trace] = ga([0 9],'gademo1eval1',[],initPop,...

[1e-6 1 1],'maxGenTerm',25,...

'normGeomSelect',[0.08],...

['arithXover'],[2],...

'nonUnifMutation',[2, 25 ,3])

% Output Arguments:

%   x    - the best solution found during the course of the

run

%   endPop       - the final population

%   bPop         - a trace of the best population

(解的变化)

%   traceInfo    - a matrix of best and means of the ga

for each generation

(种群平均值的变化)

%

% Input Arguments:

%   bounds - a matrix of upper and lower bounds

on the variables

%   evalFN  - the name of the evaluation .m function

%   evalOps

- options to pass to the evaluation function ([NULL])

%   startPop   - a matrix of solutions that can be initialized

%              from initialize.m

%   opts   - [epsilon, prob_ops ,display]

change required to consider two solutions

different, prob_ops 0 if you want to apply the

%            genetic operators probabilisticly to each solution,

1 if you are supplying a deterministic number

of operator  applications and display is 1 to output

progress 0 for  quiet. ([1e-6 1 0])

%   termFN  - name of the .m termination function

(['maxGenTerm'])

%   termOps  - options string to be passed to the termination

function  ([100]).

%   selectFN     - name of the .m selection function

(['normGeomSelect'])

%   selectOpts   - options string to be passed to select after

%                  select(pop,#,opts) ([0.08])

%   xOverFNS     - a string containing blank seperated names

of Xover.m files (['arithXover heuristicXover

simpleXover'])

%   xOverOps     - A matrix of options to pass to Xover.m files

with the first column being the number of that

xOver to perform similiarly for mutation

([2 0;2 3;2 0])

%   mutFNs       - a string containing blank seperated names of

mutation.m files (['boundaryMutation

multiNonUnifMutation ...

%                           nonUnifMutation unifMutation'])

%   mutOps       - A matrix of options to pass to Xover.m files

with the first column being the number of that

xOver to perform similiarly for mutation

([4 0 0;6 100 3;4 100 3;4 0 0])

02

2.求sin(x) 在0到3.14之间的最大值.

function [sol, val] = sin1(sol,options)

x=sol(1);

val =sin(x);

initPop=initializega(10,[0, 3.14],'sin1',[],[1e-6,1]);

[x, endPop,bpop,trace] = ga([0 3.14],'sin1',[],initPop,...

[1e-6 1 1],'maxGenTerm',25,...

'normGeomSelect',[0.08],...

['arithXover'],[2],...

'nonUnifMutation',[2, 25 ,3])

03

3. binaryExample.m

二元函数例子 二进制编码

方式1 >> binaryExample

方式2

function [sol,val] = gaMichEval(sol,options)

val = 21.5 + sol(1) * sin(4*pi*sol(1)) + sol(2)*sin(20*pi*sol(2));

%%%%%%%%%%%%%%

global bounds

% Setting the seed back to the beginning for comparison sake

rand('seed',0)

% Crossover Operators

xFns = 'simpleXover';

xOpts = [.4];

% Mutation Operators

mFns = 'binaryMutation';

mOpts = [0.005];

% Termination Operators

termFns = 'maxGenTerm';

termOps = [200]; % 200 Generations

% Selection Function

selectFn = 'roulette'

selectOps = [];

% Evaluation Function

evalFn = 'gaMichEval';

evalOps = [];

% Bounds on the variables

bounds = [-3, 12.1; 4.1, 5.8];

% GA Options [epsilon float/binar display]

gaOpts=[1e-6 0 1];

% Generate an intialize population of size 20

startPop = initializega(20,bounds,'gaMichEval',[],[1e-6 0]);

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,...

termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);

% x is the best solution found

x

% endPop is the ending population

endPop

% trace is a trace of the best value and average value of generations

trace

clf

plot(trace(:,1),trace(:,2));

hold on

plot(trace(:,1),trace(:,3));

% Lets increase the population size by running the defaults

%

rand('seed',0)

termOps=[100];

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts,termFns,termOps,...

selectFn,selectOps);

% x is the best solution found

x

% endPop is the ending population

%endPop

% trace is a trace of the best value and average value of generations

%trace

% Plot the best over time

clf

plot(trace(:,1),trace(:,2));

% Add the average to the graph

hold on

plot(trace(:,1),trace(:,3));

04

4. floatExample.m

二元函数例子 浮点编码

global bounds

% Setting the seed to the same for binary

rand('seed',156789)

% Crossover Operators

xFns = 'arithXover heuristicXover simpleXover';

xOpts = [1 0; 1 3; 1 0];

% Mutation Operators

mFns = 'boundaryMutation multiNonUnifMutation nonUnifMutation unifMutation';

mOpts = [2 0 0;3 200 3;2 200 3;2 0 0];

% Termination Operators

termFns = 'maxGenTerm';

termOps = [200]; % 200 Generations

% Selection Function

selectFn = 'normGeomSelect';

selectOps = [0.08];

% Evaluation Function

evalFn = 'gaMichEval';

evalOps = [];

% Bounds on the variables

bounds = [-3 12.1; 4.1 5.8];

% GA Options [epsilon float/binar display]

gaOpts=[1e-6 1 1];

% Generate an intialize population of size 20

startPop = initializega(20,bounds,'gaMichEval',[1e-6 1])

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,startPop,gaOpts,...

termFns,termOps,selectFn,selectOps,xFns,xOpts,mFns,mOpts);

% x is the best solution found

x

% endPop is the ending population

endPop

% bestPop is the best solution tracked over generations

bestPop

% Plot the best over time

clf

plot(trace(:,1),trace(:,2));

% Add the average to the graph

hold on

plot(trace(:,1),trace(:,3));

% Lets increase the population size by running the defaults

[x endPop bestPop trace]=ga(bounds,evalFn,evalOps,[],gaOpts);

% x is the best solution found

x

% endPop is the ending population

endPop

% bestPop is the best solution tracked over generations

bestPop

% Plot the best over time

clf

plot(trace(:,1),trace(:,2));

% Add the average to the graph

hold on

plot(trace(:,1),trace(:,3));

05

5. 求解非线性规划问题

max f(x)

s.t. g_i(x)<=0,i=1,...,m

h_i(x)=0,i=m+1,...,n

x∈Ω

& 拒绝策略

& 修复策略

& 改进遗传算子策略

& 惩罚策略

e.g. min  f(x)=(x₁-2)²+(x₂-1)²

s.t. g₁(x)=x₁-2x2+1>=0

g₂(x)=x₁²/4-x₂²+1>=0

分析：取加法形式的适值函数：

val(x)=f(x)+p(x)

惩罚函数 p(x)由两部分组成，可变乘法因子和

违反约束乘法，其表达式如下：

其中r_i是约束i的可变惩罚系数。

步骤如下：

function [sol,eval]=f552(sol,options)

x1=sol(1);

x2=sol(2);

r1=0.1;

r2=0.8;

%约束条件

g1=x1-2*x2+1;

g2=x1.^2/4-x2.^2+1;

%加惩罚项的适值

if (g1>=0)&(g2>=0)

eval=(x1-2).^2+(x2-1).^2;

else

eval=(x1-2).^2+(x2-1).^2+r1*g1+r2*g2;

end

eval=-eval;

%%%%%%%%%%%%%%%%%%%%%%%%%%%

%维数n=2

%设置参数边界

bounds = ones(2,1)*[-1 1];%??????

%遗传算法优化

[p,endPop,bestSols,trace]=ga(bounds,'f552');

p

%性能跟踪

plot(trace(:,1),trace(:,3),'b-')

hold on

plot(trace(:,1),trace(:,2),'r-')

xlabel('Generation');

ylabel('Fittness');

legend('解的变化','种群平均值的变化');

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