CONOPT由GAMS分发,似乎是GRG的一个成熟实现,但并非免费(尽管演示可能足够您使用)。
Alglib提供了非线性Levenberg-Marquardt算法的实现here,可以选择GPL /商业许可证。
以下是使用alglib的示例代码:
#include "stdafx.h"
#include <iostream>
#include <cmath>
#include "optimization.h"
using namespace std;
double fn(double a1, double a2, double a3, double x, double A, double B)
{
return A * exp(-x*(a1*B*B+a2*B+a3));
}
struct problem
{
double *m_a1s;
double *m_a2s;
double *m_a3s;
double *m_xs;
double *m_ys;
int m_n;
problem(double *a1s, double *a2s, double *a3s, double *xs, double *ys, int n)
: m_a1s(a1s), m_a2s(a2s), m_a3s(a3s), m_xs(xs), m_ys(ys), m_n(n)
{
}
void fn_vec(const alglib::real_1d_array &c_var, alglib::real_1d_array &fi, void *ptr)
{
double sum = 0.0;
for(int i = 0; i < m_n; ++i)
{
double yhat = fn(m_a1s[i], m_a2s[i], m_a3s[i], m_xs[i], c_var[0], c_var[1]);
double err_sq = (m_ys[i] - yhat) * (m_ys[i] - yhat);
sum += err_sq;
}
fi[0] = sum;
}
};
problem *g_p;
void fn_vec(const alglib::real_1d_array &c_var, alglib::real_1d_array &fi, void *ptr)
{
g_p->fn_vec(c_var, fi, ptr);
}
int main()
{
cout << "Testing non-linear optimizer..." << endl;
int n = 5;
double a1s[] = {2.42, 4.78, 7.25, 9.55, 11.54};
double a2s[] = {4.25, 5.27, 6.33, 7.32, 8.18};
double a3s[] = {3.94, 4.05, 4.17, 4.28, 4.37};
double xs[] = {0.024, 0.036, 0.048, 0.06, 0.072};
double ys[] = {80, 70, 50, 40, 45};
double initial[] = {150, 1.75};
double ss_init = 0.0;
cout << "Initial problem:" << endl;
for(int i = 0; i < n; ++i)
{
double yhat = fn(a1s[i], a2s[i], a3s[i], xs[i], initial[0], initial[1]);
double err_sq = (ys[i] - yhat) * (ys[i] - yhat);
ss_init += err_sq;
cout << a1s[i] << "\t" << a2s[i] << "\t" << a3s[i] << "\t"
<< xs[i] << "\t" << ys[i] << "\t" << yhat << "\t" << err_sq << endl;
}
cout << "Error: " << ss_init << endl;
problem p(a1s, a2s, a3s, xs, ys, n);
g_p = &p;
alglib::real_1d_array x = "[150.0, 1.75]";
double epsg = 0.00000001;
double epsf = 0;
double epsx = 0;
alglib::ae_int_t maxits = 0;
alglib::minlmstate state;
alglib::minlmreport report;
alglib::minlmcreatev(2, x, 0.0001, state);
alglib::minlmsetcond(state, epsg, epsf, epsx, maxits);
alglib::minlmoptimize(state, fn_vec);
alglib::minlmresults(state, x, report);
cout << "Results:" << endl;
cout << report.terminationtype << endl;
cout << x.tostring(2).c_str() << endl;
double ss_end = 0.0;
for(int i = 0; i < n; ++i)
{
double yhat = fn(a1s[i], a2s[i], a3s[i], xs[i], x[0], x[1]);
double err_sq = (ys[i] - yhat) * (ys[i] - yhat);
ss_end += err_sq;
cout << a1s[i] << "\t" << a2s[i] << "\t" << a3s[i] << "\t"
<< xs[i] << "\t" << ys[i] << "\t" << yhat << "\t" << err_sq << endl;
}
cout << "Error: " << ss_end << endl;
return 0;
}
这是示例输出:
./opt
Testing non-linear optimizer...
Initial problem:
2.42 4.25 3.94 0.024 80 95.5553 241.968
4.78 5.27 4.05 0.036 70 54.9174 227.485
7.25 6.33 4.17 0.048 50 24.8537 632.338
9.55 7.32 4.28 0.06 40 9.3038 942.257
11.54 8.18 4.37 0.072 45 3.06714 1758.36
Error: 3802.41
Results:
2
[92.22,0.57]
2.42 4.25 3.94 0.024 80 77.6579 5.48528
4.78 5.27 4.05 0.036 70 67.599 5.76475
7.25 6.33 4.17 0.048 50 56.6216 43.8456
9.55 7.32 4.28 0.06 40 46.0026 36.0314
11.54 8.18 4.37 0.072 45 36.6279 70.0922
Error: 161.219