c - C 中的广义约简梯度算法

Question

我正在从事一些科学项目，我需要用于非线性优化的广义约化梯度算法的 C 语言实现。是否有任何图书馆或只是一段代码？或者，请为非线性多变量问题提出任何其他解决方案。我希望使用 4 个自变量和 2 个常量构建一个优化模型:该模型是非线性的。 我已经使用广义缩减梯度 (GRG) 使用 Microsoft Excel 的求解器检查是否可以完美地求解这个模型，但我需要用 C 语言来进行模拟。

这是我的 excel 解决方案: http://speedy.sh/SEdZj/eof-cs-rest.xlsm 我使用 Microsoft Excel Solver 和 GRG 算法来搜索 SS 的最小值，输出是 Const_a 和 Const_b 的值。

Answer 1

由 GAMS 分发的 CONOPT 似乎是 GRG 的既定实现，但不是免费的(尽管演示可能对您来说已经足够)。

Alglib 实现了非线性 Levenberg-Marquardt 算法 here，并且已获得 GPL/商业许可。

下面使用 alglib 的示例代码:

/*
 * Simple optimiser example
 *
 * nl_opt.cpp
 *
 * Compile with eg 'g++ -I../tools/alglib/src ../tools/alglib/src/ap.cpp ../tools/alglib/src/alglibinternal.cpp ../tools/alglib/src/linalg.cpp ../tools/alglib/src/alglibmisc.cpp ../tools/alglib/src/solvers.cpp ../tools/alglib/src/optimization.cpp nl_opt.cpp -o opt'
 *
 */

#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;

    // create problem
    problem p(a1s, a2s, a3s, xs, ys, n);
    g_p = &p;

    // setup solver
    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);

    // optimize
    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;
}

这给出了示例输出:

./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