模拟退火算法

模拟退火算法从某一较高的温度出发，这个温度称为初始温度，伴随着温度参数的不断下降，算法中的解趋于稳定，但是，可能这样的稳定解是一个局部最优解，此时，模拟退火算法中会以一定的概率跳出这样的局部最优解，以寻找目标函数的全局最优解。如上图中所示，若此时寻找到了A点处的解，模拟退火算法会以一定的概率跳出这个解，如跳到了D点重新寻找，这样在一定程度上增加了寻找到全局最优解的可能性。

模拟退火算法流程

模拟退火算法C++实现

//simulated annealing//F(x) = 6x^7+8x^6+7x^3+5x^2-xy//x在0~100之间，输入y，求F最小值#include <iostream>#include <time.h>#include <math.h>#define ITERS 10  //迭代次数#define T 100  //温度上界 #define Tmin 1e-8  //温度下界#define delta 0.98#define INF 1e99  //double范围正负1.7e308using namespace std;double random(){double a = rand()*1.0 / RAND_MAX;  //随机数生成在0~1之间if (rand() & 1)   //x&1结果不是0就是1a *= -1;  //结果在-1~1之间return a;}double F(double x, double y){return 6 * pow(x, 7) + 8 * pow(x, 6) + 7 * pow(x, 3) + 5 * pow(x, 2) - x*y;}double SA(double y){double result = INF;double result_new = INF;double t = T;srand(time(NULL));double x[ITERS];for (int i = 0; i < ITERS; i++)x[i] = fabs(random()) * 100;  //初始化数组while (t>Tmin){for (int i = 0; i < ITERS; i++){result = F(x[i], y);  //计算目标函数double x_new = x[i] + random()*t;  //扰动在邻域内产生新解if (x_new>=0 && x_new<=100){result_new = F(x_new, y);  //新解的目标函数值if (result_new - result < 0) {x[i] = x_new;  //一定接受下降}else{double p = exp(-(result_new - result) / t);if (fabs(random()) < p)x[i] = x_new;  //以概率接受新解}}}t *= delta;}for (int i = 0; i < ITERS; i++){if (result > F(x[i], y))result = F(x[i], y);}return result;}int main(){double y;cout << "input y:";cin >> y;cout << endl;cout << "result: " << SA(y) << endl;return 0;}