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

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?

## 1 Answer

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.