What is a "regular point" for KKT, and how does it relate to Slater's condition?

Question

In proposition 3.3.1 of Non-Linear Programming by Berteskas, it is written that if $X^*$ is a regular point and minimizer of the constrained optimization problem, then there will exist unique $(\lambda^*,\mu^*)$ that will satisfy the KKT condition.

I have the following question in that regard:

1. What do we mean by regular point? Is it necessary that a convex optimization problem will satisfy the regularity condition?
2. I understand that it is not necessary for a convex optimization problem to satisfy the KKT condition. But if Slater's condition is satisfied, then KKT is satisfied. So what is the difference between Slater's condition and regularity condition?

Answer 1

Let your feasible set be defined by $$ f_i(x) \leq 0, \quad h_j(x) = 0, \text{for $i = 1, \dots, m$ and $j = 1, \dots, k$}. $$ where all functions involved are differentiable. Fix a point $x$ and denote the set of "active" inequality constraints by $$ \mathcal{I}(x) := \{ i \mid f_i(x) = 0 \}. $$

We say $x$ is regular if the collection of vectors below is linearly independent: $$ \{ \nabla h_j(x), \; \nabla f_i(x) \mid j = 1, \dots, k, \; i \in \mathcal{I}(x) \}. $$

This is also known in the literature as LICQ (linear independence constraint qualification).

In general, LICQ is a stronger condition than Slater's condition, as alluded to in this math.SE post.