If I have a convex optimisation problem for an engineering application of the standard form: $$ \begin{equation*} \begin{aligned} \min_{x} \quad & f(x)\\ \textrm{s.t.} \quad & g_i(x) \leq 0 \\ & h_j(x) = 0 \\ \end{aligned} \end{equation*} $$ with $f(x), g_i(x)$ being convex functions and $h_j(x)$ being linear functions.
And now I want to verify that Slater's conditon holds: If there exists a $x^*$ relative interior point of the convex set such that $g_i(x^*) < 0\ \forall i$ and $h_j(x^*) = 0\ \forall j$.
Can I just iterate over random point, for example, in a python script and as soon as one point would fulfill all convex strict inequalities and linear equations I have verified that Slater's condition is given and strong duality holds?