NMinimize with list as variable starting point

Question

FindMinimum has a very convenient feature:

If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.

I do not find a similar feature for NMinimize. Is there a simple work around?

Answer 1

Just reading the documentation for NMinimize tells you that this feature is not present. Furthermore, NMinimize does not take "starting point for a variable", so that is necessarily so.

Now if what you want is to find the $\vec{x}$ that minimizes $f(\vec{x})$, for example for dimension 5, and using Norm as ans example function

f = Norm;
With[{xx = Array[x, 5]},
 NMinimize[f[xx], xx]
 ]

Or you could define something like this

NVectorMinimize[f_, x_, n_] := Block[
  {
   xx = Array[Unique[], n],
   fmin,
   vec,
   func = Function[x, f][xx]
   },
  {fmin, vec} = NMinimize[func, xx];
  {fmin, x -> (xx /. vec)}
  ]


NVectorMinimize[f[x], x, 4]
(* {1.9299*10^-11, x -> {-2.5935*10^-12, -1.0298*10^-11, -9.0777*10^-13, -1.6089*10^-11}} *)