Representing an optimization program as a quadratic program with linear constraints.

Question

I am trying to convert this minimization problem:

$$ \text{minimize}_{x \in \mathbb{R}^N, z \in \mathbb{R}}. \sum_{m=1}^M \text{max} (\mathbf{a}_m^T x, z) + \tau \Vert x \Vert_2 $$

to a quadratic program with linear constraints. How can I go about this problem? Thanks.

Answer 1

Introduce a new variable $y_m$ to represent the $\max$ and impose linear constraints \begin{align} y_m &\ge \mathbf{a}_m^T x \\ y_m &\ge z \\ \end{align}