Graph Of Convex Sets at Every Scale

\begin{aligned} \min~& \sum_{e \in \mathcal{E}}y_{e}f_{e}(x_{u},x_{v})\\ \text{subject to }& y_{v}x_{v} \in y_{v}\mathcal{X}_{v} ~\forall v \in \mathcal{V} \\ & y_{e}(x_{u}, x_{v}) \in y_{e}\mathcal{X}_{e} ~ \forall (u,v) \in \mathcal{E} \\ & y \text{ encodes a path },~~ y\in \{0,1\} \end{aligned}

\begin{aligned} \min~& \sum_{e \in \mathcal{E}}c_{e}^{T}(z_{e_{u}},z_{e_{v}}, y_{e}) \\ \text{subject to }& A_{v}z_{v}-b_{v} y_{v} \in \mathcal{K}_{v} ~\forall v \in \mathcal{V} \\ & A_{e}(z_{e_{u}},z_{e_{v}}) - b_{e}y_{e} \in \mathcal{K}_{e} ~ \forall (u,v) \in \mathcal{E} \\ & A_{\mathcal{P}}y-b_{\mathcal{P}} \geq 0,~~ y\in \{0,1\}\ \end{aligned}

\begin{aligned} \min~& g^{T}z \\ \text{subject to }~& Tx - s \in \mathcal{K} \end{aligned}