Question

I am working on some science project and I need the C language implementation of Generalized Reduced Gradient algorithm for non-linear optimization. Is there any library or just a piece of code for that? Or, please suggest any other solution for non-linear multivariable problems. I am looking to build an optimization model using 4 independent variables and 2 constants: the model is nonlinear.
I have checked with Microsoft Excel’s Solver using generalized reduced gradient (GRG) is solving this model perfectly, but I need this in the C language for my simulations.

Here is my excel solution:
http://speedy.sh/SEdZj/eof-cs-rest.xlsm
I have used Microsoft Excel Solver with GRG algorithm to search for a minimum value of SS and the output are values of Const_a and Const_b.

Answer 1

CONOPT as distributed by GAMS seems an established implementation of GRG, but is not free (although demo may be sufficient for you).

Alglib has an implementation of non-linear Levenberg-Marquardt algorithm here and is GPL / commercial licensed.

Sample code using alglib below:

/*
 * 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;
}

This gives sample output:

./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