We are given a matrix $D \in \mathbb{Z}^{C \times C}$ of non-negative entries, an integer $k \geq 1$ and we need to maximize the quadratic form $x^T D x$ under some simple constraints. For all variables we have the condition $0 \leq x_i \leq U_i$ where $U_i$ is some upper bound. Some $x_i$-s are even fixed in the sense that $x_i = U_i$. Finally, the sum of the variables must be $k$, i.e. $\sum_{i} x_i = k$.
My question is whether this quadratic program can be maximized, either fractionally or integrally, in time $f(C) \cdot \text{poly}(k)$ where $f$ can be an arbitrary function. So something like $C^{C^C} \cdot k^4$ would be allowed but $k^C$ would not be allowed.
Well, to be more precise, in the case that an optimal fractional solution is sought the answer might be irrational... so assume for that case we are given some $\varepsilon > 0$ and we would like to find an $(1-\varepsilon)$-approximation of the maximizer in time $f(C,\varepsilon) \cdot \text{poly}(k)$.
I suspect that at least some partial answer should be known but I am not at all an expert and did not find anything directly applicable in the literature. Any hints are appreciated, even if they require further assumptions to be applicable. Thank you!
