For vectors, $\beta, c \in \mathbb{R}^d$, $e, y \in \mathbb{R}^n$, and matrix $B \in \mathbb{R}^{d \times n}$, I am trying to solve the following problem.
\begin{equation*} \min _{\beta \geq 0} \max_{y_{1}, y_{2}}\Bigg\{\max\left\{e^{\top} y_{1},-e^{\top} y_{1}, e^{\top} y_{2},-e^{\top} y_{2}\right\}+\beta^{\top}\left(c-\frac{1}{2} \sum_{j=1}^{2} B y_{j}\right)\Bigg\} \end{equation*}
I am not sure how to combine the terms inside the maximum operator with the y terms outside the maximum and reformulate this as a single $\min$ problem. Would appreciate some help.
Define $$ t = \max_{y_{1}, y_{2}}\left\{e^{\top} y_{1},-e^{\top} y_{1}, e^{\top} y_{2},-e^{\top} y_{2}\right\} $$ This will allow you to rewrite the original optimization problem as \begin{align} \min _{\beta \geq 0,y_1,y_2} &~~t ~+~\beta^{\top}\left(c-\frac{1}{2} \sum_{j=1}^{2} B y_{j}\right) \\ \text{s.t.}~~~& e^Ty_i \leq t~,~ -e^Ty_i \leq t ~~\forall i\in\{1,2\} \end{align} In terms of your optimization variables $(\beta,y_1,y_2,t)$, this is a quadratic optimization problem with linear constraints. You should be able to rewrite this optimization in the standard form of a quadratic program (see here) as \begin{align} \min_{x} ~~&x^TQx + r^Tx \\ \text{s.t} &~~ Ax \leq b \end{align} You will notice that $Q$ is a function of $B$ and couple of other matrices. If you are in luck due to the specific nature of your problem, $Q$ might be positive sem-definite which in turn makes it a convex problem. Otherwise, in general, this is NP-hard.