The theory of optimal transport considers the problem of transporting utilities distributed acoording to a probability measure $\mu$ on $X$ to "targets" distributed according to a probability measure $\nu$ on $Y$. In Kantorovich's formulation, this is done by looking for a transport plan $\pi$, meaning a probability measure on $X\times Y$, minimizing $$\int c(x,y)\pi(dx,dy)$$ for a cost function $c(x,y)$, interpreted as the cost of transporting something from $x\in X$ to $y\in Y$. Such a transport plan has to satisfy $\pi\circ p_X^{-1}=\mu$ and $\pi\circ p_Y^{-1}=\nu$, where $p_X,p_Y$ denote the projections onto $X,Y$.
Now consider the simple case where $X=\{x_1,\dots,x_n\}$ and $Y=\{y_1,\dots,y_m\}$ are discrete. Then any plan $\pi$ on $X\times Y$ is also discrete and in particular the Kantorovich problem is given as a linear optimization problem: $$\begin{align*}\min \sum_{i=1}^n\sum_{j=1}^m &c(x_i,y_j)\pi(x_i,y_j) \\ \text{s.t.} \sum_{i=1}^n\pi_{i,j}&=\nu_j, \hspace{1cm} \forall j=1,\dots,m \\ \sum_{j=1}^m\pi_{i,j}&=\mu_i,\hspace{1cm} \forall i=1,\dots,n \\ \pi_{i,j}&\geq 0\end{align*}$$
This in turn means that we can interpret a certain subset of linear optimization problems as optimal transport problems in Kantorovich's formulation.
Question: Is there a similar interpretation for (a subclass of) convex optimization problems from the viewpoint of optimal transport?
Here a convex optimization problem is meant to be an optimization problem of the form $$\begin{align*}\min &f(x) \\ g(x)&\preceq 0 \\ h(x)&=0 \\ x&\in C\end{align*}$$ where $f:\mathbb R^n\rightarrow \mathbb R$ and $g:\mathbb R^n\rightarrow \mathbb R^m$ are convex, $h:\mathbb R^n\rightarrow \mathbb R^d$ is affine and $C\subseteq \mathbb R^n$ is a convex set. ($v\preceq w$ is meant as $v_i\leq w_i$ for all $i$)