Briefly, have the following problem: \begin{equation} \sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\ s.t.\\\\ A \bar x \leq b \end{equation} where $ F( \bar x ) $ is a linear function, $a_i \gt 0$, $n$ is huge comparing to the size of $x$.

It is possible to write an equal Quadratic Programming problem, such as

$$ \sum_{i=0}^n a_i \ ( G_i )^2 \rightarrow min \\\\ s.t. \\\\ G_i \geq {\bf 0}, \quad i = 0..n \\\\ G_i \geq F_i( \bar x ) \quad i = 0..n \\\\ A \bar x \leq b $$

which can be solved very efficiently with an appropriate numerical method.

Unfortunately in my particular case such conversion doesn't work: it adds a lot of new restrictions, and that appropriate numerical method doesn't converge.

I tried to figure out another equal QPP, which adds fewer new constraints, but nothing came across my mind. Is there another way?

Edit: I need some time to apply both methods to my particular problem. I'll try to report on result as soon as I can.