Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance operators w.r.t the probability measure $P$. Let $P_0$ be a fixed probability measure.

Question 1: How to rewrite the problem $$\sup_{P \ll P_0}\mathbb E_P[Z]+\mathbb V_{P_0}(dP/dP_0), $$ as a 1D optimization problem $\min_{c \in \mathbb R}J(c,P_0)$ involving the reference measure $P_0$ ?

Question 2: In general, given a convex function $\varphi$ with $\varphi(z) \ge \varphi(1)=\varphi'(1) = 0$ for all $z \in \mathbb R$ and $\varphi(z) = +\infty$ for $z < 0$, how to rewrite the problem

$$\sup_{P \ll P_0}\mathbb E_P[Z]+\mathbb E_{P_0}[\varphi(dP/dP_0)], $$ as a 1D optimization problem $\min_{c \in \mathbb R}J(c,P_0;\varphi)$ involving the reference measure $P_0$ and the potential $\varphi$?

Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.

Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance operators w.r.t the probability measure $P$. Let $P_0$ be a fixed probability measure.

Question 1: How to rewrite the problem $$\sup_{P \ll P_0}\mathbb E_P[Z]+\mathbb V_{P_0}(dP/dP_0), $$ as a 1D optimization problem $\min_{c \in \mathbb R}J(c,P_0)$ involving the reference measure $P_0$ ?

Question 2: In general, given a convex function $\varphi$ with $\varphi(z) \ge \varphi(1)=\varphi'(1) = 0$ for all $z \in \mathbb R$ and $\varphi(z) = +\infty$ for $z < 0$, how to rewrite the problem

$$\sup_{P \ll P_0}\mathbb E_P[Z]+\mathbb E_{P_0}[\varphi(dP/dP_0)], $$ as a 1D optimization problem $\min_{c \in \mathbb R}J(c,P_0;\varphi)$ involving the reference measure $P_0$ and the potential $\varphi$?

Observation: I the sought-for representation should be obtainable via the envelope theorem, but I'm not sure how to get it.