(Convex) Reformulation of a program

Question

Given $\{(x_i,y_i)\in \mathbb{R}^d\times \mathbb{R}\}_{i=1}^n$, consider the the following program: \begin{eqnarray*} \mathrm{min}_{\{\hat{y}_i \in \mathbb{R}\}_{i=1}^n,\{g_k \in \mathbb{R}^d\}_{k=1}^m} & \sum_{i=1}^n (y_i-\hat{y}_i)^2\\ \mathrm{s.t. }& \hat{y}_i \geq \min_{k=1,\cdots,m}\max_{j=1,\cdots,n} \big(\hat{y}_j+g_k^T(x_i-x_j)),\forall i. \end{eqnarray*} The background for this program is trying to fit the data with a piecewise linear convex function with at most $m$ linear pieces (in dimension $1$). The constraint is basically saying that for fixed $i$, there is an subgradient $g$ such that the hyperplane property holds. Is there a way to reformulate this program into at least a convex one?