It is well-known that convex optimization problem is defined as a problem of minimizing convex function (or maximizing concave function) over a convex feasible set which is usually written as (in standard form)
$\min_x \; f(x)$
s.t. $g_i(x) \leq 0, \; i = 1, \cdots, m$
where $f(x)$ is convex, and $g_i(x)$ is convex. Here, I wonder if $g_i(x)$ should be always convex for convex optimization problems. For instance, only quasi-convexity of $g_i(x)$ seems to be sufficient for generating convex set.
For example, consider an optimization problem
$\min_{x \geq 0} \; x^2$
s.t. $ \sqrt{x} \leq 2$
It can be seen that $\sqrt{x}$ is not convex, but the feasible set of above problem is convex. Is the above problem convex or not? Also, can I apply standard convex optimization techniques such as KKT conditions or duality to find its global optimal solution?