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Let $g_1(x):=(x_1-1)^2+(x_2-1)^2-2\leq 0, g_2(x):=(x_1-1)^2+(x_2+1)^2-2\leq 0, g_3(x):=-x_1\leq 0$ be given constraints. Show that the Mangasarian-Fromovitz Constraint Qualification (MFCQ) is satisfied in $\bar x=(0,0)$ but the Linear Independence Constraint Qualification (LICQ) is not satisfied.

All three constraints are active. I determined $$\nabla g_1(\bar x)=(-1,0)^T,\nabla g_2(\bar x)=(-2,-2)^T , \nabla g_3(\bar x)=(-2,2)^T.$$

Choose $d=(2,1)^T$ to satisfy $\nabla g_i(x)^Td < 0\ \forall i=1,2,3$. So (MFCQ) is satisfied.

BUT: $\nabla g_i(x)$ are linear independent $\forall i=1,2,3$, so (LICQ) is also satisfied. Where is my mistake?

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$\nabla g_3(\bar x) = (1/4) \nabla g_1(\bar x) + (1/4) \nabla g_2(\bar x)$. So the active constraint gradients are not independent.

Even without finding that relation, it must be the case that the active constraint gradients are not independent if the number of active constraints (3 in this case) is greater than the optimization problem dimension (2 in this case). That is because the rank of a matrix (active constraint gradients in this case) can't exceed the lesser of number of rows and columns.

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  • $\begingroup$ Oh yes, makes sense. Thank you!! $\endgroup$
    – Uhmm
    Jan 16, 2023 at 15:26

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