How do you usually describe the following problem and how to solve it ?
\begin{equation*} \begin{aligned} & \underset{x,y}{\min} & & f(x,y) + z, \\ & \text{s.t.} & & z = \underset{w}{\min}\quad g(x,y,w). \end{aligned} \end{equation*} Here, $f(x,y)$ is linear in $x$ and convex in $y$. $z$ is linear in $w$. Also, it is impossible to solve the problem as, \begin{equation*} \underset{x,y,w}{\min} \quad f(x,y) + g(x,y,w). \end{equation*}