I'm reading the paper https://arxiv.org/pdf/1905.10812.pdf where strongly convex approximations to Brenier potentials are approximated.
Let $\mathcal{E}$ be a partition of $\mathbb{R}^{d}$ and $ 0\leq l\leq L.$ For $\mu, \nu\in \mathcal{P}_{2}(\mathbb{R}^{d})$, we call $f_{\star}$ a $\mathcal{E}$-locally $L$-smooth $\ell$-strongly convex nearest Brenier SSNB potential between $\mu$ and $\nu$ if
$$ f_{\star}\in\arg\min_{f\in \mathcal{F}_{\ell,L,\mathcal{E}}}W_{2}(\nabla f_{\#}\mu, \nu)\ . $$
In the paper, they also show how to estimate an SSNB potential at a selection of points.
The $n$ values $u_i = f(x_i)$ and gradients $z_i=\nabla f(x_i)$ of a SSNB potential $f\in \mathcal{F}_{\ell,L,\mathcal{E}}$ can be recovered as:
$$ \min_{{z_1,\ldots,z_n}\in \mathbb{R}^d; u\in \mathbb{R}^n} W_2^2(\sum_{i=1}^n a_i \delta_{z_i},\nu) = \min_{P\in \mathcal{U}(a,b)} \sum_{i,j} P_{ij}\left\| z_i - y_j \right\|^2 $$
s.t. $$\forall k\leq K, \forall i,j \in I_k, \\ u_i \geq u_j + \langle z_j,x_i-x_j \rangle + \frac{1}{2(1-\ell / L)}(\frac{1}{L} \left\|z_i - z_j\right\|^2 + \ell \left\|x_i - x_j \right\|^2 - 2 \frac{\ell}{L} \langle z_j - z_i,x_j - x_i \rangle).$$
Moreover, for $x\in E_k,\, v=f(x)\, \textrm{ and } g=\nabla f(x)$ can be recovered as:
$$ \min_{v\in \mathbb{R},g\in \mathbb{R}^d} v \text{s.t.} \forall i\in I_k,\, v\geq u_i + \langle{z_i, x-x_i\rangle} + \frac{1}{2(1-\ell/L)}(\frac{1}{L} \left\|g-z_i\right\|^2 + \ell \left\|x-x_i\right\|^2 - 2\frac{\ell}{L}\langle{z_i-g,x_i-x\rangle}).$$
How can I analyze the behavior of $f_{*}$ ? In particular, I want to calculate the semi-dual objective $S(f_{*})$ and compare it to $S(f^*)$ where $f^* \in \arg \min_{f\in \mathcal{C}} W_2(\nabla f_{\#}\mu,\nu)$ where $\mathcal{C}$ is the set of convex functions. In other words, is there a way to control $\left|S(f^{*})-S(f_*)\right|$ where $f^{*}$ is the solution to a convex optimization problem?
