# On approximation of maximization of quadratic function over a convex set

Given a convex function $f : \mathbb{R}^n \to [0,\infty)$. Let $L = \left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1 \right\rbrace$ be it's sub-level set and suppose that $L$ is not empty as well as having a volume of at least $\epsilon > 0$.

Let $E$ be an ellipsoid, such that it's partially contained in $L$, and is represented by a positive semidefinite matrix $G \in \mathbb{R}^{n \times n}$.

We aim in approximating the following:

$$\max_{x \in L} x^T G x$$

Is there any paper which addresses such problem or have given an approximation towards solving the problem?