matlab中遗传算法求最优解

ゝ一世哀愁。 2022-05-15 12:19 441阅读 0赞

首先什么是遗传算法：[一个非常好的理解遗传算法的例子 强烈推荐入门][Link 1]

> 遗传[算法][Link 2]的手工模拟计算示例
> 
> 为更好地理解遗传算法的运算过程，下面用手工计算来简单地模拟遗传算法的各  
>     个主要执行步骤。  
>     
>      例：求下述二元函数的最大值：
> 
> ![200812202481225480.jpg][]
> 
>     (1) 个体编码  
>            遗传算法的运算对象是表示个体的符号串，所以必须把变量 x1, x2 编码为一种  
>        符号串。本题中，用无符号二进制整数来表示。  
>            因 x1, x2 为 0 ~ 7之间的整数，所以分别用3位无符号二进制整数来表示，将它  
>        们连接在一起所组成的6位无符号二进制数就形成了个体的基因型，表示一个可  
>        行解。  
>            例如，基因型 X＝101110 所对应的表现型是：x＝\[ 5，6 \]。  
>            个体的表现型x和基因型X之间可通过编码和解码程序相互转换。
> 
> (2) 初始群体的产生  
>           遗传算法是对群体进行的进化操作，需要给其淮备一些表示起始搜索点的初始  
>       群体数据。  
>          本例中，群体规模的大小取为4，即群体由4个个体组成，每个个体可通过随机  
>      方法产生。  
>           如：011101，101011，011100，111001  
>            
>  (3) 适应度汁算  
>           遗传算法中以个体适应度的大小来评定各个个体的优劣程度，从而决定其遗传  
>        机会的大小。  
>           本例中，目标函数总取非负值，并且是以求函数最大值为优化目标，故可直接  
>        利用目标函数值作为个体的适应度。
> 
>  (4)  选择运算  
>           选择运算(或称为复制运算)把当前群体中适应度较高的个体按某种规则或模型遗传到下一代群体中。一般要求适应度较高的个体将有更多的机会遗传到下一代  
>       群体中。                     
> 本例中，我们采用与适应度成正比的概率来确定各个个体复制到下一代群体中  
>      的数量。其具体操作过程是：  
>          •  先计算出群体中所有个体的适应度的总和  fi  ( i=1.2,…,M );  
>          •  其次计算出每个个体的相对适应度的大小 fi / fi ，它即为每个个体被遗传  
>              到下一代群体中的概率，  
>          •  每个概率值组成一个区域，全部概率值之和为1；  
>          •  最后再产生一个0到1之间的随机数，依据该随机数出现在上述哪一个概率区  
>              域内来确定各个个体被选中的次数。
> 
> ![200812202485936734.jpg][]
> 
> (5)  交叉运算  
>         交叉运算是遗传算法中产生新个体的主要操作过程，它以某一概率相互交换某  
>     两个个体之间的部分染色体。  
>        本例采用单点交叉的方法，其具体操作过程是：  
>        • 先对群体进行随机配对；  
>        • 其次随机设置交叉点位置；  
>        • 最后再相互交换配对染色体之间的部分基因。
> 
> ![200812202493093964.jpg][]
> 
> (6)  变异运算  
>          变异运算是对个体的某一个或某一些基因座上的基因值按某一较小的概率进  
>      行改变，它也是产生新个体的一种操作方法。  
>         本例中，我们采用基本位变异的方法来进行变异运算，其具体操作过程是：  
>         • 首先确定出各个个体的基因变异位置，下表所示为随机产生的变异点位置，  
>           其中的数字表示变异点设置在该基因座处；  
>         • 然后依照某一概率将变异点的原有基因值取反。
> 
> ![200812202495948037.jpg][]
> 
> 对群体P(t)进行一轮选择、交叉、变异运算之后可得到新一代的群体p(t+1)。
> 
> ![200812202502724016.jpg][]
> 
> 从上表中可以看出，群体经过一代进化之后，其适应度的最大值、平均值都得  
>     到了明显的改进。事实上，这里已经找到了最佳个体“111111”。         
> \[注意\]        
>         需要说明的是，表中有些栏的数据是随机产生的。这里为了更好地说明问题，  
>    我们特意选择了一些较好的数值以便能够得到较好的结果，而在实际运算过程中  
>    有可能需要一定的循环次数才能达到这个最优结果。  
> ![20081220251181281.jpg][]

大概知道了什么叫遗传算法，那如何通过matlab的工具箱进行使用？

[数学学院遗传算法一份ppt][ppt]

提取密码：fz87

> 遗传算法的工具箱有很多种 ：  
> Genetic Algorithm Tool  简称GATOOL->GADS
> 
> 可以以命令行方式调用遗传算法函数ga

> help ga  
>  ga    Constrained optimization using genetic algorithm.  
>     ga attempts to solve problems of the following forms:  
>         min F(X)  subject to:  A\*X  <= B, Aeq\*X  = Beq (linear constraints)  
>          X                     C(X) <= 0, Ceq(X) = 0 (nonlinear constraints)  
>                                LB <= X <= UB  
>                                X(i) integer, where i is in the index  
>                                vector INTCON (integer constraints)  
>    
>     Note: If INTCON is not empty, then no equality constraints are allowed.  
>     That is:-  
>     \* Aeq and Beq must be empty  
>     \* Ceq returned from NONLCON must be empty  
>    
>     X = ga(FITNESSFCN,NVARS) finds a local unconstrained minimum X to the  
>     FITNESSFCN using ga. NVARS is the dimension (number of design  
>     variables) of the FITNESSFCN. FITNESSFCN accepts a vector X of size  
>     1-by-NVARS, and returns a scalar evaluated at X.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b) finds a local minimum X to the function  
>     FITNESSFCN, subject to the linear inequalities A\*X <= B. Linear  
>     constraints are not satisfied when the PopulationType option is set to  
>     'bitString' or 'custom'. See the documentation for details.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local minimum X to the  
>     function FITNESSFCN, subject to the linear equalities Aeq\*X = beq as  
>     well as A\*X <= B. (Set A=\[\] and B=\[\] if no inequalities exist.) Linear  
>     constraints are not satisfied when the PopulationType option is set to  
>     'bitString' or 'custom'. See the documentation for details.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub) defines a set of lower and  
>     upper bounds on the design variables, X, so that a solution is found in  
>     the range lb <= X <= ub. Use empty matrices for lb and ub if no bounds  
>     exist. Set lb(i) = -Inf if X(i) is unbounded below;  set ub(i) = Inf if  
>     X(i) is unbounded above. Linear constraints are not satisfied when the  
>     PopulationType option is set to 'bitString' or 'custom'. See the  
>     documentation for details.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON) subjects the  
>     minimization to the constraints defined in NONLCON. The function  
>     NONLCON accepts X and returns the vectors C and Ceq, representing the  
>     nonlinear inequalities and equalities respectively. ga minimizes  
>     FITNESSFCN such that C(X)<=0 and Ceq(X)=0. (Set lb=\[\] and/or ub=\[\] if  
>     no bounds exist.) Nonlinear constraints are not satisfied when the  
>     PopulationType option is set to 'bitString' or 'custom'. See the  
>     documentation for details.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON,options) minimizes  
>     with the default optimization parameters replaced by values in the  
>     structure OPTIONS. OPTIONS can be created with the GAOPTIMSET function.  
>     See GAOPTIMSET for details.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b,\[\],\[\],lb,ub,NONLCON,INTCON) requires that  
>     the variables listed in INTCON take integer values. Note that ga does  
>     not solve problems with integer and equality constraints. Pass empty  
>     matrices for the Aeq and beq inputs if INTCON is not empty.  
>    
>     X = ga(FITNESSFCN,NVARS,A,b,\[\],\[\],lb,ub,NONLCON,INTCON,options)  
>     minimizes with integer constraints and the default optimization  
>     parameters replaced by values in the structure OPTIONS. OPTIONS can be  
>     created with the GAOPTIMSET function. See GAOPTIMSET for details.  
>    
>     X = ga(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a structure  
>     that has the following fields:  
>         fitnessfcn: <Fitness function>  
>              nvars: <Number of design variables>  
>              Aineq: <A matrix for inequality constraints>  
>              bineq: <b vector for inequality constraints>  
>                Aeq: <Aeq matrix for equality constraints>  
>                beq: <beq vector for equality constraints>  
>                 lb: <Lower bound on X>  
>                 ub: <Upper bound on X>  
>            nonlcon: <Nonlinear constraint function>  
>             intcon: <Index vector for integer variables>  
>            options: <Options structure created with GAOPTIMSET>  
>           rngstate: <State of the random number generator>  
>    
>     \[X,FVAL\] = ga(FITNESSFCN, ...) returns FVAL, the value of the fitness  
>     function FITNESSFCN at the solution X.  
>    
>     \[X,FVAL,EXITFLAG\] = ga(FITNESSFCN, ...) returns EXITFLAG which  
>     describes the exit condition of ga. Possible values of EXITFLAG and the  
>     corresponding exit conditions are  
>    
>       1 Average change in value of the fitness function over  
>          options.StallGenLimit generations less than options.TolFun and  
>          constraint violation less than options.TolCon.  
>       3 The value of the fitness function did not change in  
>          options.StallGenLimit generations and constraint violation less  
>          than options.TolCon.  
>       4 Magnitude of step smaller than machine precision and constraint  
>          violation less than options.TolCon. This exit condition applies  
>          only to nonlinear constraints.  
>       5 Fitness limit reached and constraint violation less than  
>          options.TolCon.  
>       0 Maximum number of generations exceeded.  
>      -1 Optimization terminated by the output or plot function.  
>      -2 No feasible point found.  
>      -4 Stall time limit exceeded.  
>      -5 Time limit exceeded.  
>    
>     \[X,FVAL,EXITFLAG,OUTPUT\] = ga(FITNESSFCN, ...) returns a  
>     structure OUTPUT with the following information:  
>               rngstate: <State of the random number generator before ga started>  
>            generations: <Total generations, excluding HybridFcn iterations>  
>              funccount: <Total function evaluations>  
>          maxconstraint: <Maximum constraint violation>, if any  
>                message: <ga termination message>  
>    
>     \[X,FVAL,EXITFLAG,OUTPUT,POPULATION\] = ga(FITNESSFCN, ...) returns the  
>     final POPULATION at termination.  
>    
>     \[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORES\] = ga(FITNESSFCN, ...) returns  
>     the SCORES of the final POPULATION.  
>    
>    
>     Example:  
>       Unconstrained minimization of 'rastriginsfcn' fitness function of  
>       numberOfVariables = 2  
>        x = ga(@rastriginsfcn,2)  
>    
>       Display plotting functions while ga minimizes  
>        options = gaoptimset('PlotFcns',...  
>          \{@gaplotbestf,@gaplotbestindiv,@gaplotexpectation,@gaplotstopping\});  
>        \[x,fval,exitflag,output\] = ga(@rastriginsfcn,2,\[\],\[\],\[\],\[\],\[\],\[\],\[\],options)  
>    
>     An example with inequality constraints and lower bounds  
>      A = \[1 1; -1 2; 2 1\];  b = \[2; 2; 3\];  lb = zeros(2,1);  
>      % Use mutation function which can handle constraints  
>      options = gaoptimset('MutationFcn',@mutationadaptfeasible);  
>      \[x,fval,exitflag\] = ga(@lincontest6,2,A,b,\[\],\[\],lb,\[\],\[\],options);  
>    
>       FITNESSFCN can also be an anonymous function:  
>          x = ga(@(x) 3\*sin(x(1))+exp(x(2)),2)  
>    
>     If FITNESSFCN or NONLCON are parameterized, you can use anonymous  
>     functions to capture the problem-dependent parameters. Suppose you want  
>     to minimize the fitness given in the function myfit, subject to the  
>     nonlinear constraint myconstr, where these two functions are  
>     parameterized by their second argument a1 and a2, respectively. Here  
>     myfit and myconstr are MATLAB file functions such as  
>    
>          function f = myfit(x,a1)  
>          f = exp(x(1))\*(4\*x(1)^2 + 2\*x(2)^2 + 4\*x(1)\*x(2) + 2\*x(2) + a1);  
>    
>     and  
>    
>          function \[c,ceq\] = myconstr(x,a2)  
>          c = \[1.5 + x(1)\*x(2) - x(1) - x(2);  
>                -x(1)\*x(2) - a2\];  
>          % No nonlinear equality constraints:  
>           ceq = \[\];  
>    
>     To optimize for specific values of a1 and a2, first assign the values  
>     to these two parameters. Then create two one-argument anonymous  
>     functions that capture the values of a1 and a2, and call myfit and  
>     myconstr with two arguments. Finally, pass these anonymous functions to  
>     ga:  
>    
>       a1 = 1; a2 = 10; % define parameters first  
>       % Mutation function for constrained minimization  
>       options = gaoptimset('MutationFcn',@mutationadaptfeasible);  
>       x = ga(@(x)myfit(x,a1),2,\[\],\[\],\[\],\[\],\[\],\[\],@(x)myconstr(x,a2),options)  
>    
>     Example: Solving a mixed-integer optimization problem  
>     An example of optimizing a function where a subset of the variables are  
>     required to be integers:  
>       
>     % Define the objective and call ga. Here variables x(2) and x(3) will  
>     % be integer.  
>     fun = @(x) (x(1) - 0.2)^2 + (x(2) - 1.7)^2 + (x(3) -5.1)^2;  
>     x = ga(fun,3,\[\],\[\],\[\],\[\],\[\],\[\],\[\],\[2 3\])  
>              
>     See also gaoptimset, FITNESSFUNCTION, gaoutputfcntemplate, patternsearch, @.
> 
>

（模模糊糊不知道咋用）下面有两个实例：

函数 ga—— 带线性约束条件

![70][]

> 适应度函数另存为：fun\_1.m在当前目录  
> function y=fun\_1(x); %x为自变量行向量  
> c=\[2 4 6\]; % c为目标函数系数向量  
> y=c\*x‘ % y为适应度函数值，求两向量的数量积
> 
>  
> 
> A=\[1 0 -1;-1 -2 0;-1 0 0\]; % A为AX<=b的系数矩阵  
> b=\[10;-1;0\]; % b为AX<=b的常数项  
> Aeq=\[0 1 1\]; %AeqX=beq的系数矩阵  
> beq=12; %AeqX=beq的常数项
> 
> >> \[x,y\]=ga(@fun\_1,3,A,b,Aeq,beq,\[\],\[\]);  
> >> \[x,y\]=ga(@fun\_1,3,A,b,Aeq,beq); % 或

函数 ga—— 带非线性约束条件

![70 1][]

这里有个要注意的地方，ga函数求取的是最小值。（参看help）

> 适应度函数另存为：fun\_2.m在当前目录  
> function y=fun\_2(x); %x为自变量行向量  
> temp=\[1 2 5\]\*x'+\[3 0 1\]\*x'.^2;  
> y=-temp; % 转换为求最小值
> 
> function \[budengshi,dengshi\]=yueshu(x);%非线性约束  
> budengshi=\[4\*x(1)^2+2\*x(2)^2+x(3)-46;  
> x(1)+x(1)^2+x(2)^2-5\*x(3)-34;  
> 3\*x(1)+2\*x(1)^2+x(2)+3\*x(3)-67\];  
> dengshi=x(1)+x(3)^2-13;
> 
> A=\[-2 -4 -6; -1 0 0; 0 1 0\]; % A为AX<=b的系数矩阵  
> b=\[-10; 0; 10\]; % b为AX<=b的常数项  
> >>\[x,y\]=ga(@fun\_2,3,A,b,\[\],\[\],\[\],\[\],@yueshu)  
> >>y=-y

至此，在matlab中优化工具箱中，利用ga函数（遗传算法），求解线性与非线性最优解的问题结束。

[Link 1]: https://blog.csdn.net/u012422446/article/details/68061932
[Link 2]: http://lib.csdn.net/base/datastructure
[200812202481225480.jpg]: /images/20220515/2b36926267284df79d06b57b91cb24a9.png
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[20081220251181281.jpg]: /images/20220515/b0245b65518b42d49cdef287e1cccc90.png
[ppt]: https://pan.baidu.com/s/1qDqzgKNKmS4boXq5a71Egw
[70]: /images/20220515/1b1894baaea14992aefdfd4d99fbce9e.png
[70 1]: /images/20220515/e2a29dec552140c2ba41136468e21928.png