Question

我正在研究一些科学项目，我需要用于非线性优化的广义简化梯度算法的C语言实现。是否有任何库或只是一段代码？或者，请为非线性多变量问题建议任何其他解决方案。我期待使用4个独立变量和2个常数构建一个优化模型：模型是非线性的。我用Microsoft Excel的Solver检查使用广义简化梯度（GRG）完美地解决了这个模型，但我需要用C语言进行模拟。

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（尽管演示可能已经足够）。

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

使用以下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;
}

这给出了样本输出：

./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

C中的广义简约梯度算法

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