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Proposition of a better convergion criterion in multimin

Hi all,

I found that the stopping criterion proposed in
gsl_multimin_test_gradient suffers from scale problems. Typically, if
you have variables of magnitude 1.0e0 and a function of magnitude 1.0e5,
it can be impossible to minimize the norm of the gradient under 1.0e-2.
Dennis and Schnabel in "Numerical Methods for Unconstrained Optimization
and Nonlinear Equations", p.160 propose another criterion:
relgrad = gradfi * xi / f,
where gradfi is the ith component of the gradient and xi the ith
variable. The criterion is ||relgrad||inf < epsabs, with ||.||inf is the
infinite norm: ||x||inf=max(|xi|).
Here is a proposition of a new routine called gsl_multimin_test_relgrad:

gsl_multimin_test_relgrad (const gsl_vector *g, const gsl_vector *x,
double f, double epsabs)
  int i;
  double relgrad;

  if (epsabs < 0.0)
      GSL_ERROR ("absolute tolerance is negative", GSL_EBADTOL);

  if (relgrad < epsabs)
      return GSL_SUCCESS;

  return GSL_CONTINUE;

Take care


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