If I add a decimal point to the first constant inside each Abs then Minimize completes in seconds and returns
{9.25992, {a -> 1., b -> 2., c -> 3., d -> 4., e -> 0.794416, f -> 1.40992}}
If I instead use NMinimize with Method->"RandomSearch" then it completes in seconds and returns
{7.76376, {a -> 1., b -> 2., c -> 3., d -> 4., e -> 3.16253, f -> 5.4614}}
How can this value be smaller than the value returned by Minimize?
If I read the documentation for Minimize under Details and Options it says
"If f and cons are linear or polynomial, Minimize will always find a global minimum."
You have Abs and rational polynomials and that is neither linear nor polynomial.
If you need a global minimum then it appears you need another function or to rewrite your problem in polynomials, and not use Sqrt when trying to avoid Abs.
The help page for Minimize also says
"If Minimize is given an expression containing approximate numbers, it automatically calls NMinimize."
That explains why the result is returned so much faster.
