= Solution
Separate inactive and active <optimization constraints>. At an interior feasible point with $h>0$, a differentiable <local minimum> must satisfy $\nabla f=0$. Retain only solutions lying in the interior.
On the boundary $h=0$, if $\nabla h\ne0$, apply the <Lagrange multiplier> condition. Using the <optimization Lagrangian> $L=f-\lambda h$, the <Karush-Kuhn-Tucker conditions> are
$$
\nabla f=\lambda\nabla h,\qquad h\geq0,\qquad
\lambda\geq0,\qquad\lambda h=0.
$$
The sign follows from feasible motion into $h>0$: the directional <derivative> along $\nabla h$ at a minimum cannot be negative. Thus a boundary candidate has $h=0$ and a nonnegative <Lagrange multiplier>. Boundary points with $\nabla h=0$ must be examined separately, because the usual constraint qualification fails there.
Solve the interior and boundary systems, check membership in the <feasible set>, and establish which candidates are minima using second-order tests or a direct global comparison. The <Karush-Kuhn-Tucker conditions> alone are not sufficient for general nonconvex <functions> or <feasible sets>. On an unbounded <feasible set>, also check whether an infimum is attained and whether escaping <sequences> can improve the objective.
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