### 37.9 Examples

This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in x and y, and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.

    int
    main (void)
    {
      size_t iter = 0;
      int status;

      const gsl_multimin_fdfminimizer_type *T;
      gsl_multimin_fdfminimizer *s;

      /* Position of the minimum (1,2), scale factors
         10,20, height 30. */
      double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };

      gsl_vector *x;
      gsl_multimin_function_fdf my_func;

      my_func.n = 2;
      my_func.f = my_f;
      my_func.df = my_df;
      my_func.fdf = my_fdf;
      my_func.params = par;

      /* Starting point, x = (5,7) */
      x = gsl_vector_alloc (2);
      gsl_vector_set (x, 0, 5.0);
      gsl_vector_set (x, 1, 7.0);

      T = gsl_multimin_fdfminimizer_conjugate_fr;
      s = gsl_multimin_fdfminimizer_alloc (T, 2);

      gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);

      do
        {
          iter++;
          status = gsl_multimin_fdfminimizer_iterate (s);

          if (status)
            break;

          status = gsl_multimin_test_gradient (s->gradient, 1e-3);

          if (status == GSL_SUCCESS)
            printf ("Minimum found at:\n");

          printf ("%5d %.5f %.5f %10.5f\n", iter,
                  gsl_vector_get (s->x, 0),
                  gsl_vector_get (s->x, 1),
                  s->f);

        }
      while (status == GSL_CONTINUE && iter < 100);

      gsl_multimin_fdfminimizer_free (s);
      gsl_vector_free (x);

      return 0;
    }

The initial step-size is chosen as 0.01, a conservative estimate in this case, and the line minimization parameter is set at 0.0001. The program terminates when the norm of the gradient has been reduced below 0.001. The output of the program is shown below,

             x       y         f
        1 4.99629 6.99072  687.84780
        2 4.98886 6.97215  683.55456
        3 4.97400 6.93501  675.01278
        4 4.94429 6.86073  658.10798
        5 4.88487 6.71217  625.01340
        6 4.76602 6.41506  561.68440
        7 4.52833 5.82083  446.46694
        8 4.05295 4.63238  261.79422
        9 3.10219 2.25548   75.49762
       10 2.85185 1.62963   67.03704
       11 2.19088 1.76182   45.31640
       12 0.86892 2.02622   30.18555
    Minimum found at:
       13 1.00000 2.00000   30.00000

Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points.

The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic. Additional iterations would be needed for a more complicated function.

Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as above.

    int
    main(void)
    {
      double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};

      const gsl_multimin_fminimizer_type *T =
        gsl_multimin_fminimizer_nmsimplex2;
      gsl_multimin_fminimizer *s = NULL;
      gsl_vector *ss, *x;
      gsl_multimin_function minex_func;

      size_t iter = 0;
      int status;
      double size;

      /* Starting point */
      x = gsl_vector_alloc (2);
      gsl_vector_set (x, 0, 5.0);
      gsl_vector_set (x, 1, 7.0);

      /* Set initial step sizes to 1 */
      ss = gsl_vector_alloc (2);
      gsl_vector_set_all (ss, 1.0);

      /* Initialize method and iterate */
      minex_func.n = 2;
      minex_func.f = my_f;
      minex_func.params = par;

      s = gsl_multimin_fminimizer_alloc (T, 2);
      gsl_multimin_fminimizer_set (s, &minex_func, x, ss);

      do
        {
          iter++;
          status = gsl_multimin_fminimizer_iterate(s);

          if (status)
            break;

          size = gsl_multimin_fminimizer_size (s);
          status = gsl_multimin_test_size (size, 1e-2);

          if (status == GSL_SUCCESS)
            {
              printf ("converged to minimum at\n");
            }

          printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f\n",
                  iter,
                  gsl_vector_get (s->x, 0),
                  gsl_vector_get (s->x, 1),
                  s->fval, size);
        }
      while (status == GSL_CONTINUE && iter < 100);

      gsl_vector_free(x);
      gsl_vector_free(ss);
      gsl_multimin_fminimizer_free (s);

      return status;
    }

The minimum search stops when the Simplex size drops to 0.01. The output is shown below.

        1  6.500e+00  5.000e+00 f() = 512.500 size = 1.130
        2  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
        3  5.250e+00  4.000e+00 f() = 290.625 size = 1.409
        4  5.500e+00  1.000e+00 f() = 252.500 size = 1.409
        5  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
        6  2.625e+00  3.500e+00 f() = 101.406 size = 1.847
        7  0.000e+00  3.000e+00 f() =  60.000 size = 1.847
        8  2.094e+00  1.875e+00 f() =  42.275 size = 1.321
        9  2.578e-01  1.906e+00 f() =  35.684 size = 1.069
       10  5.879e-01  2.445e+00 f() =  35.664 size = 0.841
       11  1.258e+00  2.025e+00 f() =  30.680 size = 0.476
       12  1.258e+00  2.025e+00 f() =  30.680 size = 0.367
       13  1.093e+00  1.849e+00 f() =  30.539 size = 0.300
       14  8.830e-01  2.004e+00 f() =  30.137 size = 0.172
       15  8.830e-01  2.004e+00 f() =  30.137 size = 0.126
       16  9.582e-01  2.060e+00 f() =  30.090 size = 0.106
       17  1.022e+00  2.004e+00 f() =  30.005 size = 0.063
       18  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
       19  1.022e+00  2.004e+00 f() =  30.005 size = 0.043
       20  1.022e+00  2.004e+00 f() =  30.005 size = 0.027
       21  1.022e+00  2.004e+00 f() =  30.005 size = 0.022
       22  9.920e-01  1.997e+00 f() =  30.001 size = 0.016
       23  9.920e-01  1.997e+00 f() =  30.001 size = 0.013
    converged to minimum at
       24  9.920e-01  1.997e+00 f() =  30.001 size = 0.008

The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins to decrease as the simplex contracts around the minimum.