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.