Re: Multidimensional minimization with V-shaped minimum

[Stephan Lorenzen]

Do I get undefined behavior if the first derivative is not continuous?
What is the worst thing that might happen? Do I have to use another
minimizer? Is BFGS a proper choice?

Here is a minimal example using the "absolute value" function and
starting at 1 (returns status GSL_ENOPROG and 1 as minimum)

#include <iostream>
#include <gsl/gsl_multimin.h>

double f(const gsl_vector* x, void* params)
{
    double xVal = gsl_vector_get(x, 0);
    if (xVal < 0) return -xVal;
    return xVal;
}
void df(const gsl_vector* x, void* params, gsl_vector* df)
{
    double xVal = gsl_vector_get(x, 0);
    if (xVal < 0) gsl_vector_set(df, 0, -1);
    else          gsl_vector_set(df, 0,  1);
}
void fdf(const gsl_vector* x, void* params, double* f, gsl_vector* df)
{
    double xVal = gsl_vector_get(x, 0);
    if (xVal < 0)
    {
        gsl_vector_set(df, 0, -1);
        *f = -xVal;
    }
    else
    {
        gsl_vector_set(df, 0, 1);
        *f = xVal;
    }
}
int main()
{
    size_t iter = 0;
    int status;
    const gsl_multimin_fdfminimizer_type *T =
gsl_multimin_fdfminimizer_vector_bfgs2;
    gsl_multimin_function_fdf func;
    func.n = 1;
    func.f = f;
    func.df = df;
    func.fdf = fdf;
    func.params = nullptr;
    gsl_multimin_fdfminimizer *m = gsl_multimin_fdfminimizer_alloc (T, 1);
    gsl_vector *x= gsl_vector_alloc (1);
    gsl_vector_set (x, 0, 1);

    gsl_multimin_fdfminimizer_set (m, &func, x, 0.01/*step size*/,
0.01/*tol*/);
    do
    {
        iter++;
        status = gsl_multimin_fdfminimizer_iterate (m);
        if (status) break;
        status = gsl_multimin_test_gradient (m->gradient, 1);
    }
    while (status == GSL_CONTINUE);

    std::cout << "Status: " << status << std::endl;
    std::cout << "xMin: " <<
gsl_vector_get(gsl_multimin_fdfminimizer_x(m), 0) << std::endl;

    gsl_multimin_fdfminimizer_free (m);
    gsl_vector_free (x);
}

On 05.02.21 17:09, Mark Galassi wrote:
> > [...] with discontinuous first derivative at the minimum. So, the
> > first derivative never comes close to zero [...] f and df [...]
> I'm curious as to why you would use the methods that require the derivative - 
> they tend to require that the function be at least continuously differentiable 
> (i.e. derivative is continuous), and sometimes even twice differentiable.
>
> But in any case, to try to understand what's happening with 
> gsl_multimin_fdfminimizer_m(), could you give us an MWE (minimum working 
> example) that exhibits this behavior?  Then we can jump onto the problem.
>
> https://en.wikipedia.org/wiki/Minimal_working_example

--
Dr. Stephan Lorenzen
Mühlenredder 35
21493 Schwarzenbek
Tel.: 0177 872 8817