Consider the set of inequalities \begin{equation} \sum_{i=1}^{n} c_{i} x_{i}^2 \geq 0, \quad \forall x \in \mathbb{R}^{n} : Ax = b, \end{equation} for some matrices $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^{m \times 1}$ ($n>1$). For which $A$ and $b$ do the inequalities imply that $c_{i} \geq 0$ for all $i$?
For example, if $A=0_{m \times n}$ and $b=0_{m \times 1}$, then the inequalities imply that $c_{i} \geq 0$ for all $i$. And if we impose a constraint on the form $x_{1}+x_{2} = 1$ we get the same result. If $m=n$ and $A$ is non-singular, then the inequality does not imply that the $c$:s are non-negative, giving us a necessary condition on $A$. Intuitively, $Ax=b$ cannot define to many constraints on $x$ or else there is not enough degrees of freedom to enforce non-negative coefficients.
Unfortunately I haven't been able to infer anything really interesting from experimenting with various special cases, so therefore I ask whether one can give some sufficient conditions that hold for a wide range of matrices and can be checked in practice?