Maximize and NMaximize in comparison

Question

If on one hand:

N@Maximize[{Sqrt[y^2 - x^2], x + y^2 == 0 && 0 < y < 2 && -y < x < y}, {x, y}]

{0.5, {x -> -0.5, y -> 0.707107}}

which is what is desired, on the other hand:

NMaximize[{Sqrt[y^2 - x^2], x + y^2 == 0 && 0 < y < 2 && -y < x < y}, {x, y}]

produces a warning:

NMaximize::nrnum: The function value 4.15623 -1.34058 I is not a real number at {x,y} = {-1.35352,0.186697}.

and a consequent incorrect result:

{0.285088, {x -> -0.681805, y -> 0.739008}}

I can't understand why this behavior. Could you explain it to me? Thank you!

Answer 1

Sqrt isn't handled properly by NMaximize, I think.

Workaround

NMaximize[{Sqrt@RealAbs[y^2 - x^2], x + y^2 == 0 && 0 < y < 2 && -y < x < y}, {x, y}]
(*{0.5, {x -> -0.5, y -> 0.707107}}*)

gives the expected result.