How do slack variables destroy convexity? Suppose we have a convex optimization problem
$$ \min_{x \in D} f(x) \\ \textrm{subject to} \quad g_{i}(x) \leq 0\\ $$
Why does transforming the problem into
$$ \min_{x \in D, s \in R^{n}} f(x) \\ \textrm{subject to} \quad g_{i}(x) + s_{i} = 0 \\ \quad s_{i} \geq 0 $$
Destroy convexity when $g_{i}(x)$ is not affine or linear?