Slater constraint qualification: Does equality constraint imply Slater fails? Does equality constraint always imply that the Slater condition fails. For example, consider the optimization problem: $$\min_{X\in S^n_+} \ tr(AX)$$ $$s.t.\ XX^T=I$$ Since we can't have $$XX^T-I \succ 0$$ and $$-XX^T+I\succ0$$ does that mean Slater fails? Under what condition, equality constraint implies Slater fails?

The Slater constraint qualification pertains to convex optimization problems. Nonlinear equality constraints are non-convex, except in trivial cases such as $x^2 = 0$. Your constraint $XX^T = I$ is non-convex, so your optimization problem is non-convex and the Slater constraint qualification does not apply.

As mentioned, the Slater constraint qualification only "restricts" nonlinear constraints, and is not applied to linear constraints (equality or inequality).