I have an optimization problem with a constraint of the following form:
\begin{align} \inf_y\quad &J(y)\\ \mbox{s.t.}\quad & y\in\mathcal{Y}\\ &\min_{j}a_j^\top y+b_j\leq \delta \end{align} where $\mathcal{Y}$ is a convex set. I understand that the second constraint is a nonconvex one. But is there no way to tackle it without using mixed integer formulation?
One way I thought of was using the epigraph form, which is incorrect in this case, as it results in the following:
\begin{align} \inf_y\quad &J(y)\\ \mbox{s.t.}\quad & y\in\mathcal{Y}\\ &t\leq \delta\\ &a_j^\top y+b_j\geq t,\;\forall j \end{align}
I would appreciate any recommendation.