Can the following bilevel optimization problem be reformulated into an equivalent linear program?
$\begin{aligned} \min _{\lambda, a} & \lambda \\ \text {s.t. } & |w|-\lambda+a w \leq 0, \quad \forall w \in [c, d]\end{aligned}$
I tried to split $w$ into $w^{+}$ and $w^{-}$; however, the resulting linear program has linearly independent inequality constraints.
Any hints will be appreciated. Thanks in advance!
I have not worked out the details, but here are two hints. First, you can linearize the constraint by noting that $|w|=\max(w,-w)$: \begin{align} w&\le\lambda-aw \\ -w&\le\lambda-aw \end{align} The other hint is to rewrite the semi-infinite linear programming problem explicitly as bilevel optimization and dualize the inner LP.