Are KKT regularity conditions always fulfilled for a single inequality?

Question

When I was studying the KKT regularity conditions it appeared to me that they would always be fulfilled for a single inequality constraint since LICQ only requires the gradients to be independent at the extreme point. Is this understanding correct?

Answer 1

Welcome to Mathematics Stackexchange! This understanding is not correct because a single gradient may be linearly dependent if it is the null vector. For example, consider the optimization problem $$\text{minimize }-\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x}$$

$$\text{subject to } \boldsymbol{x}^{\mathrm{T}} \boldsymbol{x}\leq 1.$$

Then, we can write the constraint as $g(x) := x_1^2+...+x_n^2-1 = 0$ and get $ \nabla g(x) = 2(x_1,\dots, x_n) ' ,$ which is only different from $0$ if $x\neq 0$. Hence, at $x=0$, the LICQ fails to hold.