This is a caveat 
on the reformulation technique in optimization.
We (partly) encounter this problem \upsilon = \min_{x > 0} -x \log(x),
which is also -\upsilon = \max_{x > 0} x \log(x).
Someone may think it is a convex maximization problem.
Regretfully, this turns out to be an illusion under the Max operator.
Indeed, the maximization problem can be reformulated as:
\max_x f(x), where
f(x) = -\infty when x \le 0,
f(x) = x \log(x) when x > 0.
Now we see that f is improper and nonconvex.