I am stuck with the following optimization problem:
$$\min_w \sum_j (\max\{0,w^Tx_j\}-b_j)^2 $$
the $x_j \in \mathbb{R}^n$ and $b_j>0$ are given and i am searching the w minimizing this. My approach was to find a constrained dual formulation. But up to now I have not found anything reasonable. I got to the following formulation:
$$\min_{w,t} \sum_j (t_j-b_j)^2 $$ s.t. $$w^Tx_j < t_j$$ $$w^Tx_j t_j \geq 0$$
Without the second constraint, when $w^Tx_j<0 \Rightarrow t_j = b_j$ at the optimum. But optimizing this formulation with the second constraint is rather horrible.
Is there a more elegant solution?
