I've the following "pseudo" Quadratic Program (QP). Suppose $x\in\mathbb{R}^n$
$$ x^{\text{opt}} = \arg\min_{x} \,\,\,\,\dfrac{1}{2}x^{\intercal} M x + \dfrac{1}{2}x^\intercal T + \dfrac{1}{2}x^\intercal F(x) $$
where $M\in\mathbb{R}^{n\times n}$, $(M + M^\intercal)$ is positive-definite, $T$ is any matrix with proper dimensions, and the mapping $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is nonlinear. I'm interested in using state-of-the-art QP solvers, which are extremely computationally efficient, but, of course, this problem is not "entirely" in QP form, given the presence of the mapping $F$.
My question is: is there any way I can treat this problem as a "perturbed" QP optimization, provided some condition in the norm of $F(x)$? I obviously don't mind if I lose some accuracy in the solution of the original problem... my application requires real-time performance.
Thanks in advance!