Set
f[a_?NumberQ, b_?NumberQ] := 5
and attempt
NMinimize[{f[a, b], a>0.1 && a<0.2 && b>0 && b<10 && Element[b,Integers], {a,b}]
The response is
NMinimize::nsol: There are no points that satisfy the constraints {}.
Yet if you try
NMinimize[{f[a, b], a>0.0 && a<0.2 && b>0 && b<10 && Element[b,Integers], {a,b}]
you get
{5., {a -> 0, b -> 6}}
Function is smooth with all derivatives.
{5., {a -> 0.15, b -> 6}}
would seem like a reasonable solution to the first request in light of the second.