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1、遗传算法求函数最优解
题目要求：
f(x1,x2) = 21.5+x1*sin(4pi*x1)+x2*sin(20pi*x2)
st：约束范围
x1:[-3.0,12.1]
x2:[4.1,5.8]
求函数在约束范围内的最大值
2、算法流程图：
3、Genetic.h文件
#ifndef _GENETIC_H_
#define _GENETIC_H_
using namespace std;
#define PI 3.14159265358979323846
//遗传算法参数，种群规模（0~100）、繁殖代数、函数变量个数、交叉概率、编译概率
# define GROUP_SCALE 50
# define MAX_GENS 500
# define N_VARS 2
# define P_MATING 0.8
# define P_MUTATION 0.15
struct Individual
{
double Xn[N_VARS]; //存放变量值
double Fitness; //适应值
double ReFitness; //适应值概率密度
double SumFitness; //累加分布，为轮盘转
};
struct X_Range
{
double Upper; //变量的上界取值
double Lower; //变量的下界取值
};
template<typename T>
T randT(T Lower, T Upper); //产生任意类型随机数函数
void crossover(int &seed);
void elitist(); //基因保留
void evaluate();
void initGroup(int &seed);
void selectBest();
void mutate(int &seed);
double r8_uniform_ab(double a, double b, int &seed);
int i4_uniform_ab(int a, int b, int &seed);
void report(int Xnration);
void selector(int &seed);
void showTime();
void Xover(int one, int two, int &seed);
#endif // !_GENETIC_H_
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4、Genetic.cpp
# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <fstream>
# include <iomanip>
# include <cmath>
# include <ctime>
# include <cstring>
#include "Genetic.h"
//申请种群内存，其中多加1个是放置上一代中最优秀个体
struct Individual Population[GROUP_SCALE + 1];
X_Range XnRange[N_VARS] = { { -3.0,12.1}, {4.1,5.8} };
//有交配权的所有父代进行交叉
void crossover(int &seed)
{
const double a = 0.0;
const double b = 1.0;
int mem;
int one;
int first = 0;
double x;
for (mem = 0; mem < GROUP_SCALE; ++mem)
{
x = randT(0.0,1.0);
//x = r8_uniform_ab(a, b, seed);//产生交配概率
if (x < P_MATING)
{
++first;
if (first % 2 == 0)//交配
{
Xover(one, mem, seed);
}
else
{
one = mem;
}
}
}
return;
}
//对最差的一代和最优的一代的处理，起到优化的目的
void elitist()
{
int i;
double best;
int best_mem;
double worst;
int worst_mem;
best = Population[0].Fitness;
worst = Population[0].Fitness;
for (i = 0; i < GROUP_SCALE - 1; ++i)
{
if (Population[i + 1].Fitness < Population[i].Fitness)
{
if (best <= Population[i].Fitness)
{
best = Population[i].Fitness;
best_mem = i;
}
if (Population[i + 1].Fitness <= worst)
{
worst = Population[i + 1].Fitness;
worst_mem = i + 1;
}
}
else
{
if (Population[i].Fitness <= worst)
{
worst = Population[i].Fitness;
worst_mem = i;
}
if (best <= Population[i + 1].Fitness)
{
best = Population[i + 1].Fitness;
best_mem = i + 1;
}
}
}
//对于当前代的最优值的处理，如果当前的最优值小于上一代则将上一代的值最优个体取代当前的最弱个体
//基因保留
if (Population[GROUP_SCALE].Fitness <= best)
{
for (i = 0; i < N_VARS; i++)
{
Population[GROUP_SCALE].Xn[i] = Population[best_mem].Xn[i];
}
Population[GROUP_SCALE].Fitness = Population[best_mem].Fitness;
}
else
{
for (i = 0; i < N_VARS; i++)
{
Population[worst_mem].Xn[i] = Population[GROUP_SCALE].Xn[i];
}
Population[worst_mem].Fitness = Population[GROUP_SCALE].Fitness;
}
return;
}
//计算适应度值
void evaluate()
{
int member;
int i;
double x[N_VARS + 1];
for (member = 0; member < GROUP_SCALE; member++)
{
for (i = 0; i < N_VARS; i++)
{
x[i + 1] = Population[member].Xn[i];
}
Population[member].Fitness = 21.5 + x[1] * sin(4 * PI*x[1]) + x[2] * sin(20 * PI*x[2]);
}
return;
}
//产生整形的随机数
int i4_uniform_ab(int a, int b, int &seed)
{
int c;
const int i4_huge = 2147483647;
int k;
float r;
int value;
if (seed == 0)
{
cerr << "\n";
cerr << "I4_UNIFORM_AB - Fatal error!\n";
cerr << " Input value of SEED = 0.\n";
exit(1);
}
//保证a小于b
if (b < a)
{
c = a;
a = b;
b = c;
}
k = seed / 127773;
seed = 16807 * (seed - k * 127773) - k * 2836;
if (seed < 0)
{
seed = seed + i4_huge;
}
r = (float)(seed)* 4.656612875E-10;
//
// Scale R to lie between A-0.5 and B+0.5.
//
r = (1.0 - r) * ((float)a - 0.5)
+ r * ((float)b + 0.5);
//
// Use rounding to convert R to an integer between A and B.
//
value = round(r);//四舍五入
//保证取值不越界
if (value < a)
{
value = a;
}
if (b < value)
{
value = b;
}
return value;
}
//初始化种群个体
void initGroup(int &seed)
{
int i;
int j;
double lbound;
double ubound;
//
// initGroup variables within the bounds
//
for (i = 0; i < N_VARS; i++)
{
//input >> lbound >> ubound;
for (j = 0; j < GROUP_SCALE; j++)
{
Population[j].Fitness = 0;
Population[j].ReFitness = 0;
Population[j].SumFitness = 0;
Population[j].Xn[i] = randT(XnRange[i].Lower, XnRange[i].Upper);
//Population[j].Xn[i] = r8_uniform_ab(XnRange[i].Lower, XnRange[i].Upper, seed);
}
}
return;
}
//挑选出最大值，保存在种群数组的最后一个位置
void selectBest()
{
int cur_best;
int mem;
int i;
cur_best = 0;
for (mem = 0; mem < GROUP_SCALE; mem++)
{
if (Population[GROUP_SCALE].Fitness < Population[mem].Fitness)
{
cur_best = mem;
Population[GROUP_SCALE].Fitness = Population[mem].Fitness;
}
}
for (i = 0; i < N_VARS; i++)
{
Population[GROUP_SCALE].Xn[i] = Population[cur_best].Xn[i];
}
return;
}
//个体变异
void mutate(int &seed)
{
const double a = 0.0;
const double b = 1.0;
int i;
int j;
double lbound;
double ubound;
double x;
for (i = 0; i < GROUP_SCALE; i++)
{
for (j = 0; j < N_VARS; j++)
{
//x = r8_uniform_ab(a, b, seed);
x = randT(a, b);//突变概率
if (x < P_MUTATION)
{
lbound = XnRange[j].Lower;
ubound = XnRange[j].Upper;
Population[i].Xn[j] = randT(lbound, ubound);
//Population[i].Xn[j] = r8_uniform_ab(lbound, ubound, seed);
}
}
}
return;
}
//模板函数，用于生成各种区间上的数据类型
template<typename T>
T randT(T Lower, T Upper)
{
return rand() / (double)RAND_MAX *(Upper - Lower) + Lower;
}
//产生小数随机数
double r8_uniform_ab(double a, double b, int &seed)
{
int i4_huge = 2147483647;
int k;
double value;
if (seed == 0)
{
cerr << "\n";
cerr << "R8_UNIFORM_AB - Fatal error!\n";
cerr << " Input value of SEED = 0.\n";
exit(1);
}
k = seed / 127773;
seed = 16807 * (seed - k * 127773) - k * 2836;
if (seed < 0)
{
seed = seed + i4_huge;
}
value = (double)(seed)* 4.656612875E-10;
value = a + (b - a) * value;
return value;
}
//输出每一代进化的结果
void report(int Xnration)
{
double avg;
double best_val;
int i;
double square_sum;
double stddev;
double sum;
double sum_square;
if (Xnration == 0)
{
cout << "\n";
cout << " Xnration Best Average Standard \n";
cout << " number value Fitness deviation \n";
cout << "\n";
}
sum = 0.0;
sum_square = 0.0;
for (i = 0; i < GROUP_SCALE; i++)
{
sum = sum + Population[i].Fitness;
sum_square = sum_square + Population[i].Fitness * Population[i].Fitness;
}
avg = sum / (double)GROUP_SCALE;
square_sum = avg * avg * GROUP_SCALE;
stddev = sqrt((sum_square - square_sum) / (GROUP_SCALE - 1));
best_val = Population[GROUP_SCALE].Fitness;
cout << " " << setw(8) << Xnration
<< " " << setw(14) << best_val
<< " " << setw(14) << avg
<< " " << setw(14) << stddev << "\n";
return;
}
//选择有交配权的父代
void selector(int &seed)
{
struct Individual NewPopulation[GROUP_SCALE + 1];//临时存放挑选的后代个体
const double a = 0.0;
const double b = 1.0;
int i;
int j;
int mem;
double p;
double sum;
sum = 0.0;
for (mem = 0; mem < GROUP_SCALE; mem++)
{
sum = sum + Population[mem].Fitness;
}
//计算概率密度
for (mem = 0; mem < GROUP_SCALE; mem++)
{
Population[mem].ReFitness = Population[mem].Fitness / sum;
}
// 计算累加分布，思想是轮盘法
Population[0].SumFitness = Population[0].ReFitness;
for (mem = 1; mem < GROUP_SCALE; mem++)
{
Population[mem].SumFitness = Population[mem - 1].SumFitness +
Population[mem].ReFitness;
}
// 选择个体为下一代繁殖，选择优秀的可能性大，这是轮盘法的奥秘之处
for (i = 0; i < GROUP_SCALE; i++)
{
p = r8_uniform_ab(a, b, seed);
if (p < Population[0].SumFitness)
{
NewPopulation[i] = Population[0];
}
else
{
for (j = 0; j < GROUP_SCALE; j++)
{
if (Population[j].SumFitness <= p && p < Population[j + 1].SumFitness)
{
NewPopulation[i] = Population[j + 1];
}
}
}
}
//更新后代个体
for (i = 0; i < GROUP_SCALE; i++)
{
Population[i] = NewPopulation[i];
}
return;
}
//显示系统时间
void showTime()
{
# define TIME_SIZE 40
static char time_buffer[TIME_SIZE];
const struct tm *tm;
size_t len;
time_t now;
now = time(NULL);
tm = localtime(&now);
len = strftime(time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm);
cout << time_buffer << "\n";
return;
# undef TIME_SIZE
}
//交叉产生子代
void Xover(int one, int two, int &seed)
{
int i;
int point;
double t;
//随机选择交叉点，这里的点是以变量的整个长度为单位
point = randT<int>(0, N_VARS - 1);
//point = i4_uniform_ab(0, N_VARS - 1, seed);
//交叉
for (i = 0; i < point; i++)
{
t = Population[one].Xn[i];
Population[one].Xn[i] = Population[two].Xn[i];
Population[two].Xn[i] = t;
}
return;
}
5、main.cpp
#include <iostream>
#include "Genetic.h"
extern Individual Population[GROUP_SCALE + 1];
int main()
{
int Xnration;
int i;
int seed = 123456789;
showTime();
initGroup(seed);
evaluate();
selectBest();
for (Xnration = 0; Xnration < MAX_GENS; Xnration++)
{
selector(seed);
crossover(seed);
mutate(seed);
report(Xnration);
evaluate();
elitist();
}
cout << "\n";
cout << " Best member after " << MAX_GENS << " Xnrations:\n";
cout << "\n";
for (i = 0; i < N_VARS; i++)
{
cout << " X(" << i + 1 << ") = " << Population[GROUP_SCALE].Xn[i] << "\n";
}
cout << "\n";
cout << " Best Fitness = " << Population[GROUP_SCALE].Fitness << "\n";
showTime();
while (1);
return 0;
}
6、运行平台VS2013，500代迭加之后结果如下：
函数f（x1,x2）在x1=11.6249和x2=5.7268取得优解值: 38.8153
7、总结
遗传算法看似随机，但背后却是非随机的，是随机中朝最优解的一种策略。其过程有做了保证朝优值的方向的策略，一个是变异，另外一个是基因保留。不断的可以保证种群朝着最优方向进化。同时遗传算法的适应度函数的设计要求通常需要满足单值、连续、非负、最大化、计算量小、通用性强，因此适应度函数的设计需要根据结合实际的问题本身的特点而定。
遗传算法设计的几个关键参数：种群规模、交配率、变异率、进化代数；这几个参数的设计对函数的最终收敛性有着关系。
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发表于 2024-6-22 09:46:18
