1 模拟退火
模拟退火算法其实是一个类似于仿生学的算法，模仿的就是物理退火的过程。
我们炼钢的时候，如果我们急速冷凝，这时候的状态是不稳定的，原子间杂乱无章的排序，能量很高。而如果我们让钢水慢慢冷凝，很缓慢的降温，那么这个时候的状态就是很稳定的，各个分子都趋向于自己能量最低的位置。而模拟退火算法，恰恰就是利用了物理退火这一过程的原理，求解一个优化目标（目标函数）的最小值。
模拟退火算法来源于固体退火原理，是一种基于概率的算法，将固体加温至充分高，再让其徐徐冷却，加温时，固体内部粒子随温升变为无序状，内能增大，而徐徐冷却时粒子渐趋有序，在每个温度都达到平衡态，最后在常温时达到基态，内能减为最小。
2 模拟退火算法
模拟退火算法（Simulated Annealing，SA）最早由Metropolis等人于 1953 年提出。1983 年Kirkpatrick等人第一次使用模拟退火算法求解组合优化问题后，它就发表在了 Science 上。直到今天，它依然被广泛使用，这篇文章将详细介绍C#的代码实现。
3 单纯形法
单纯形法是针对求解线性规划问题的一个算法，这个名称里的'单纯形'是代数拓扑里的一个概念
实际问题当中变量x是多维的，约束条件也会比示例多的多，这就需要一个一劳永逸的算法能通过计算机来获得正解，单纯形法就是这样的一个算法。单纯形法最早由 George Dantzig于1947年提出，单纯形法对于求解线性规划问题是具有跨时代意义的，其实不仅仅是针对线性规划，对于非线性规划问题在求解的过程中也大量依赖单纯形法，George Dantzig本人也被称为'线性规划之父'，他是好莱坞电影《心灵捕手》的原型。
4 C#源程序
using System;
namespace Legalsoft.Truffer
{
/// <summary>
/// 模拟退火下山单纯形极小化
/// Downhill simplex minimization with simulated annealing
/// </summary>
public class Amebsa
{
public RealValueFun funk;
public Ranq1 ran;
private int mpts { get; set; }
private int ndim { get; set; }
private double[] pb { get; set; }
private double[] y { get; set; }
private double[,] p { get; set; }
private double ftol { get; }
private double yb { get; set; }
private double tt { get; set; }
public Amebsa(double[] point, double del, RealValueFun funkk, double ftoll)
{
this.funk = funkk;
this.ran = new Ranq1(1234);
this.ftol = ftoll;
this.yb = double.MaxValue;
this.ndim = point.Length;
this.pb = new double[ndim];
this.mpts = ndim + 1;
this.y = new double[mpts];
this.p = new double[mpts, ndim];
for (int i = 0; i < mpts; i++)
{
for (int j = 0; j < ndim; j++)
{
p[i, j] = point[j];
}
if (i != 0)
{
p[i, i - 1] += del;
}
}
inity();
}
public Amebsa(double[] point, double[] dels, RealValueFun funkk, double ftoll)
{
this.funk = funkk;
this.ran = new Ranq1(1234);
this.ftol = ftoll;
this.yb = double.MaxValue;
this.ndim = point.Length;
this.pb = new double[ndim];
this.mpts = ndim + 1;
this.y = new double[mpts];
this.p = new double[mpts, ndim];
for (int i = 0; i < mpts; i++)
{
for (int j = 0; j < ndim; j++)
{
p[i, j] = point[j];
}
if (i != 0)
{
p[i, i - 1] += dels[i - 1];
}
}
inity();
}
public Amebsa(double[,] pp, RealValueFun funkk, double ftoll)
{
this.funk = funkk;
this.ran = new Ranq1(1234);
this.ftol = ftoll;
this.yb = double.MaxValue;
this.ndim = pp.GetLength(1);
this.pb = new double[ndim];
this.mpts = pp.GetLength(0);
this.y = new double[mpts];
this.p = pp;
inity();
}
public void inity()
{
double[] x = new double[ndim];
for (int i = 0; i < mpts; i++)
{
for (int j = 0; j < ndim; j++)
{
x[j] = p[i, j];
}
y[i] = funk.funk(x);
}
}
public bool anneal(ref int iter, double temperature)
{
double[] psum = new double[ndim];
tt = -temperature;
get_psum(p, psum);
for (; ; )
{
int ilo = 0;
int ihi = 1;
double ylo = y[0] + tt * Math.Log(ran.doub());
double ynhi = ylo;
double yhi = y[1] + tt * Math.Log(ran.doub());
if (ylo > yhi)
{
ihi = 0;
ilo = 1;
ynhi = yhi;
yhi = ylo;
ylo = ynhi;
}
for (int i = 3; i <= mpts; i++)
{
double yt = y[i - 1] + tt * Math.Log(ran.doub());
if (yt <= ylo)
{
ilo = i - 1;
ylo = yt;
}
if (yt > yhi)
{
ynhi = yhi;
ihi = i - 1;
yhi = yt;
}
else if (yt > ynhi)
{
ynhi = yt;
}
}
double rtol = 2.0 * Math.Abs(yhi - ylo) / (Math.Abs(yhi) + Math.Abs(ylo));
if (rtol < ftol || iter < 0)
{
Globals.SWAP(ref y[0], ref y[ilo]);
for (int n = 0; n < ndim; n++)
{
Globals.SWAP(ref p[0, n], ref p[ilo, n]);
}
if (rtol < ftol)
{
return true;
}
else
{
return false;
}
}
iter -= 2;
double ytry = amotsa(p, y, psum, ihi, ref yhi, -1.0);
if (ytry <= ylo)
{
ytry = amotsa(p, y, psum, ihi, ref yhi, 2.0);
}
else if (ytry >= ynhi)
{
double ysave = yhi;
ytry = amotsa(p, y, psum, ihi, ref yhi, 0.5);
if (ytry >= ysave)
{
for (int i = 0; i < mpts; i++)
{
if (i != ilo)
{
for (int j = 0; j < ndim; j++)
{
psum[j] = 0.5 * (p[i, j] + p[ilo, j]);
p[i, j] = psum[j];
}
y[i] = funk.funk(psum);
}
}
iter -= ndim;
get_psum(p, psum);
}
}
else
{
++iter;
}
}
}
public void get_psum(double[,] p, double[] psum)
{
for (int n = 0; n < ndim; n++)
{
double sum = 0.0;
for (int m = 0; m < mpts; m++)
{
sum += p[m, n];
}
psum[n] = sum;
}
}
public double amotsa(double[,] p, double[] y, double[] psum, int ihi, ref double yhi, double fac)
{
double[] ptry = new double[ndim];
double fac1 = (1.0 - fac) / ndim;
double fac2 = fac1 - fac;
for (int j = 0; j < ndim; j++)
{
ptry[j] = psum[j] * fac1 - p[ihi, j] * fac2;
}
double ytry = funk.funk(ptry);
if (ytry <= yb)
{
for (int j = 0; j < ndim; j++)
{
pb[j] = ptry[j];
}
yb = ytry;
}
double yflu = ytry - tt * Math.Log(ran.doub());
if (yflu < yhi)
{
y[ihi] = ytry;
yhi = yflu;
for (int j = 0; j < ndim; j++)
{
psum[j] += ptry[j] - p[ihi, j];
p[ihi, j] = ptry[j];
}
}
return yflu;
}
}
}