I want to write the following problem as a linear program:
$$minimize_x \sum_{i=0}^m{h_w(a_i^Tx-b_i)}$$ $$ h_w(u) = \left\{ \begin{array}{ll} |x| & \quad |u| \leq w \\ w+(|x|-w)^2 & \quad |u| > w \end{array} \right. $$
Here is my attempt:
first, write it in the epigraph form (and because each term is positive we can separate terms): $$minimize_{x,t}\;1^Tt$$ $${h_w(a_i^Tx-b_i)} \leq t_i\:\forall i=0,..,m$$ Which could lead to: $$minimize_{x,t}\;1^Tt$$ $$ -t_i \leq{x} \leq t_i\:\forall i=0,..,m$$ $$ (|x|-w)^2 \leq {(t_i-w)}\:\forall i=0,..,m$$
I don't know how to make it simpler. Is there anyway to find out a problem could convert to an LP or not?