【模拟退火入门】1：Country Meow 2：经典问题:旅行商问题 3：求多项式最值

学习博客：

https://www.cnblogs.com/zwfymqz/archive/2018/04/28/8966901.html

https://blog.csdn.net/baimafujinji/article/details/6474567

https://www.cnblogs.com/caijiaming/p/11674182.html

https://www.cnblogs.com/cieusy/p/9871095.html

https://blog.csdn.net/qq_18343569/article/details/50300147

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模拟退火

定义：

         模拟退火算法（Simulate Anneal，SA）是一种通用概率演算法，用来在一个大的搜寻空间内找寻命题的最优解

变量：

         T:当前温度，从高温到低温，也就是从不稳定状态到稳定状态，代表算法进行到了什么程度

          ΔT:每次的温度变化率，一般取0.95-0.99，模拟退火过程，上次的温度×ΔT=当前的温度

          

算法流程：

①：枚举温度T

②：计算下一步的状态

③：若下一步的状态比当前状态优或者满足进行转移的条件就进行转移

④：降温

 

 

 

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1：Country Meow

题意：

         求三维坐标下包围n个点的最小的球的半径

 

题解：

        每次找到一个距离当前点最远的点，作为可转移的下一个状态，

         概率Δp=(据当前位置最远的点-当前点的位置）/初始状态

 

代码

//#include
#include
#include
#include
#include
#include
#include
#include

 

 

---

 

旅行商问题：

还不会，直接放网上找的代码了，当模板用了

#include 
#include 
#include 
using namespace std;

#define MAX_CITY_NUM 100
#define T0 2000
#define T 1e-5
#define ALPHA 0.98
#define L 20000

struct path{
    int route[MAX_CITY_NUM];
    double dis;
};

double w[MAX_CITY_NUM][MAX_CITY_NUM];
int num;
double s = 0;
path p0;

void Init_City()
{
    cout << "Please input the number of cities:" << endl;
    cin >> num;
    for (int i = 0; i < num; ++i)
        for (int j = 0; j < num; ++j)
            cin >> w[i][j];
}

void Init_path()
{
    p0.dis = 0;
    for (int i = 0; i < num; ++i)
    {
        p0.route[i] = i;
        if (i != num - 1)
            p0.dis += w[i][i + 1];
    }
    p0.dis += w[num - 1][0];
}

path generate(path p)
{
    int x = 0, y = 0;
    while (x == y)
    {
        x = (int)(num * (rand() / (RAND_MAX + 1.0)));
        y = (int)(num * (rand() / (RAND_MAX + 1.0)));
    }
    path gen = p;
    int tmp;
    tmp = gen.route[x];
    gen.route[x] = gen.route[y];
    gen.route[y] = tmp;

    gen.dis = 0;
    for (int i = 0; i < num - 1; ++i)
        gen.dis += w[gen.route[i]][gen.route[i + 1]];
    gen.dis += w[gen.route[num - 1]][gen.route[0]];
    return gen;
}



void TSP_SA()
{
    double t = T0;
    srand(time(NULL));
    path cur = p0;
    path next = p0;
    int bChange;
    while (t > T)
    {
        bChange = 0;
        for (int i = 0; i < L; ++i)
        {
            next = generate(cur);
            double df = next.dis - cur.dis;
            if (df <= 0)
            {
                cur = next;
                bChange = 1;
            }
            else
            {
                double rndp = rand() / (RAND_MAX + 1.0);
                double eps = exp(-df / t);
                if (eps > rndp && eps < 1)
                {
                    cur = next;
                    bChange = 1;
                }
            }

        }

        if (cur.dis < p0.dis)
            p0 = cur;

        t *= ALPHA;
        if (!bChange)
            ++s;
        else
            s = 0;

        if (s == 2)
            break;
    }
}

int main()
{
    Init_City();
    Init_path();
    TSP_SA();

    cout << "The best path is:" << endl;
    for (int i = 0; i < num; ++i)
            cout << p0.route[i] << " -> ";
    cout << p0.route[0] << endl;
    cout << "The distance of the best path is: " << p0.dis << endl;
    system("pause");
    return 0;
}

 

 

---

 

3：多项式最值问题：

还是不会，回头填坑。。

 

#include
#include
#include
#include
#include
#include
#include
using namespace std;

#define ITERS 10
#define T 100
#define eps 1e-8
#define delta 0.98
#define INF 1e99

double x[ITERS];
int judge(double x, double y)
{
	if (fabs(x - y) < eps) return 0;
	return x < y ? -1 : 1;
}

double random()
{
	double x = rand()*1.0 / RAND_MAX;
	if (rand() & 1) x *= -1;
	return x;
}

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;
}
void init()
{
	for (int i = 0; i < ITERS; i++)
	{
		x[i] = fabs(random()) * 100;
	}
}

double sa(double y)
{
	double ans = INF;
	double t = T;
	while (t < eps)
	{
		for (int i = 0; i < ITERS; i++)
		{
			double tmp = F(x[i], y);
			for (int j = 0; j < ITERS; j++)
			{
				double _x = x[i] + random()*t;
				if (judge(_x, 0) >= 0 && judge(_x, 100) <= 0)
				{
					double f = F(_x, y);
					if (tmp>f)
						x[i] = _x;
				}
			}
		}
		t = t*delta;
	}
	for (int i = 0; i < ITERS; i++)
	{
		ans = min(ans, F(x[i], y));
	}
	return ans;
}
int main()
{
	int t;
	cin >> t;
	while (t--)
	{
		double y;
		cin >> y;
		srand(time(NULL));
		init();
		cout << sa(y) << endl;
	}
	return 0;
}