Multidimensional Minimization: GNU GSL C++ error code 27 for - iteration is not making progress towards solution

Question

I am trying to find the minimum of the function using GNU scientific library, package Multidimensional Minimization. The method I am using is Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm which is implemented in function

gsl_multimin_fdfminimizer_vector_bfgs2

Unfortunately, after the first iteration with

gsl_multimin_fdfminimizer_iterate

I receive an error with code 27: iteration is not making progress towards solution.

I tried several initial points and different tolerance/step combinations, but still the same error pops up. Could you please tell me what could be the culprit here?

The function I am trying to minimize is Gupta-Potential for Aluminum 13 atoms cluster.

I also noticed that forces calculated on the first step are the SAME as gradients after first interation. Should it be like that? Shouldn't step come into the picture somehow? I use

gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.001, 1e-10);

And function (Gupta potential) does NOT change after iteration (I print it before and after) though gradients are big.

// HERE IS MY GUPTA POTENTIAL//

double
      my_f (const gsl_vector *v, void *params)
     {
       double *p = (double *)params;
        int ii,jj;
        double vv = 0;
        double sum1 = 0;
        double sum2 = 0;

          for(ii=1;ii<=NN;ii++){
          X[ii]=gsl_vector_get (v, 3*ii-3);
          Y[ii]=gsl_vector_get (v, 3*ii-2);
          Z[ii]=gsl_vector_get (v, 3*ii-1 );
         }

        for(ii = 1; ii <= NN; ii++){
                sum1 = 0;
                for(jj = 1; jj <= NN; jj++){

                        if(jj!=ii){
                                sum1 += uA*exp(0 - uP*((rr(ii,jj)/uR0) - 1.));
                        }
                }
                sum2 = 0;
                for(jj = 1; jj <= NN; jj++){
                        if(jj!=ii){
                                sum2 += uXi*uXi*exp(0 - 2*uQ*((rr(ii,jj)/uR0) - 1.));
                        }
                }
                vv += sum1 - sqrt(sum2);
        }
        return vv;
 }

// HERE ARE MY GRADIENT FUNCTIONS//

double dVdr(int ii, int jj){
        double vr;
        double sum1 = 0;
        double sum2 = 0;
        int kk;

        vr = 0. - 2.*uA*(uP/uR0)*exp(0. - uP*((rr(ii,jj)/uR0) - 1.));

        for(kk = 1; kk <= NN; kk++){
                if(kk != ii){
                        sum1 += uXi*uXi*exp(0 - 2*uQ*((rr(ii,kk)/uR0) - 1.));
                }
                if(kk != jj){
                        sum2 += uXi*uXi*exp(0 - 2*uQ*((rr(kk,jj)/uR0) - 1.));
                }
        }

        vr -= 0.5*uXi*uXi*(0. - 2.*uQ/uR0)*exp(0. - 2.*uQ*((rr(ii,jj)/uR0) - 1.))*( (1./sqrt(sum1)) + (1./sqrt(sum2)) );

        return vr;
}

//-----------------------------------------------
 void Force(int jj, double & fx, double & fy, double & fz){
        int mm;
        double dxx, dyy, dzz, r;
        fx = 0;
        fy = 0;
        fz = 0;
   for(mm = 1; mm <= NN; mm++){

    if(mm != jj){
                        dxx = X[mm] - X[jj];
                        dyy = Y[mm] - Y[jj];
                        dzz = Z[mm] - Z[jj];

     r = sqrt(dxx*dxx + dyy*dyy + dzz*dzz);
     fx += dVdr(mm,jj)*(dxx)/r;
     fy += dVdr(mm,jj)*(dyy)/r;
     fz += dVdr(mm,jj)*(dzz)/r;
   }
   }
}
//--------------------------------------------
 void
     my_df (const gsl_vector *v, void *params,
            gsl_vector *df)
     {
       int ii;
       double fx,fy,fz;
       double *p = (double *)params;

          for(ii=1;ii<=NN;ii++){

          X[ii]=gsl_vector_get (v, 3*ii-3);
          Y[ii]=gsl_vector_get (v, 3*ii-2);
          Z[ii]=gsl_vector_get (v, 3*ii-1 );
         }

          for(ii=1;ii<=NN;ii++){
          Force(ii,fx,fy,fz);
          gsl_vector_set(df, 3*ii-3, fx );
          gsl_vector_set(df, 3*ii-2, fy );
          gsl_vector_set(df, 3*ii-1, fz );
         }
     }
//----------------------------------

Answer 1

Most likely, the programmed gradient does not match the programmed objective.

You can check this by choosing a few random $x, p$ (including your starting point as $x$, if you provided one) and printing for $s_i=10^{20-i}$, $i=0:40$ the quotients

$q_i = (f(x+ s_ip)-f(x))/(s_ig(x)^Tp)$,

where $g$ is your gradient. The sequence of $q$'s must converge to 1, well, almost, as sooner or later numerical instability sets in and forces the $q_i$ to be zero (and somewhat random one or two steps earlier).

If this is not the case,either the routine for $f$ or that for $g$ is in error. You can check the individual components of $g$ by picking $p$ as the $(0,1)$-unit vectors in turn.