I want to convert the following problem into SOCP form:
$minimize \quad$ $x^TAx+a^Tx$:
$subject$ $to \quad$ $Bx \leq b$
The approach I am taking is introducing new variables, $u$ and $v$, such that:
$u=x^TAx$ and $v=a^Tx$.
Then, our problem becomes:
$minimize \quad$ $u+v$:
$subject$ $to \quad$ $Bx \leq b$, $u \leq x^TAx$ and $v \leq a^Tx$.
Among these 3 constraints, the first and the third are linear, so we can let them be as they are. Only the second constraint needs to be converted into 2-norm form somehow, which I am not able to get how to do. Could anyone kindly tell me how to do that conversion? Would I need to introduce more new variables, or does it involve using some formula or trick that I can't seem to think of?