Reference for elementary result in optimization 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?
 A: $\renewcommand{\Re}{\mathbb{R}}$
First, you do not need twice differentiability of even continuity for this to hold, and you do not need to assume that $F$ is  a convex function. You do have to assume however that for each $q$ the set $\Phi = \{(z,q): F(z,q)\leq 0\}$ is convex.
This result is often referred to as convexity of the inf-projection and it is stated as follows:
Let $f:\Re^n\times \Re^m\to \Re\cup \{\infty\}$ ($f$ can take the value $+\infty$ so as to encode constraints) and define $p(u) = \inf_x f(x,u)$ and $P(u) = \mathrm{argmin}_x f(x,u)$. Then $p$ is convex on $\Re^m$ and $P$ is convex-valued (or it returns $\varnothing$).
This is stated as Proposition 2.22 in: R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Springer, 2009.
In your case, you have the following problem:
$$\begin{aligned}
\mathbb{P}(q): \mathrm{minimise}\ U(z)\\
\text{s.t. } F(z,q) \leq 0
\end{aligned}$$
This can be equivalently written as 
$$\begin{aligned}
\mathbb{P}(q): \mathrm{minimise}\ f(z,w)
\end{aligned}$$
where $f(z,q) = U(z) + \delta_\Phi(z,q)$,
where $\delta_\Phi(z,q)=0$ if $(z,q)\in \Phi$, that is $F(z,q)\leq 0$ and $\delta_\Phi(z,q)=+\infty$ otherwise. This is called the indicator of $\Phi$. Function $f$ is then convex and you can apply the above result.
