Reference for elementary result in optimization

Question

Let $U(\mathbf{z})$ be a convex, twice differentiable function, and $F(\mathbf{z},\mathbf{q})$ be convex and twice differentiable separately in $\mathbf{z}$ and $\mathbf{q}$. Consider the problem of minimizing $U(\mathbf{z})$ subject to the constraint $F(\mathbf{z},\mathbf{q}) \le 0$. Assume that a solution exists for all $\mathbf{q}$, giving the minimum value as a function of $\mathbf{q}$. Unless I'm missing something, it's an easy result that this function is convex in $\mathbf{q}$. Does this result have a name? Surprisingly, I haven't seen it in some cursory googling. My guess is that the hypotheses above are too strict, so is there a more general statement?