Fix a homogeneous semi-algebraic set $S \subset \mathbb{R}^{n}$, by which I mean that the set $S$ is defined by inequalities $f_{1},...,f_{k} \geq 0$, where $f_{1},...,f_{k}$ are homogeneous polynomials on $\mathbb{R}^{n}$. Given linear functions $g_{1},...,g_{l}$, I'd like to test if they are non-negative on $S$.
Is this possible in principle? Is there an algorithm with a name for doing this? Is there software for checking this?
EDIT: Is it possible to determine if the linear functions $g_{1},...,g_{l}$ are positive when $S$ is the space of semi-positive definite matrices subject to a linear constraint on the entries?