Example satisfying Mangasarian-Fromovitz CQ but not LICQ

Question

On Wikipedia's page for the KKT conditions, it is stated that Mangasarian-Fromovitz constraint qualification (MFCQ) is weaker than linear independent constraint qualification (LICQ). What is a counter-example to the claim MFCQ $\Rightarrow$ LICQ?

Answer 1

Consider the following minimization problem

\begin{eqnarray} \min &&\quad f(x) ~&= -x &\\ \text{s.t.} &&\quad g_1(x) & =x &\le 0\\ &&\quad g_2(x) & = 2x &\le 0 \end{eqnarray}

which attains the unique global minimum for $x^* = 0$. Note that both constraints are active in this point.

By definition, $\nabla g_1(x^*) = 1$ and $\nabla g_2(x^*) = 2$.

MFCQ

The gradients of the equality constraints are linearly independent at $x^*$ and there exists a vector $d \in \mathbb{R}^n$ such that $\nabla g_i(x^*)^\top d < 0$ for all active inequality constraints and $\nabla h_j(x^*)^\top d = 0$ for all equality constraints.

In this case, we can choose $d = -1$ to obtain $\nabla g_i(x^*)^\top d < 0$ for all $i \in \{1,2\}$. We have no equality constraints. It follows that MFCQ holds.

LICQ

The gradients of the active inequality constraints and the gradients of the equality constraints are linearly independent at $x^*$.

At $x^*$, both constraints are active and their gradients are clearly not independent. It follows that LICQ does not hold.

We conclude that MFCQ $\Rightarrow$ LICQ is false.