Most statements of Constraint Qualification I have found in the literature mention a locally "locally optimal solution" of the problem: $$ \begin{cases} \min f(x) \\ \text{s.t.}\\ g_i(x)\leq 0 \end{cases}$$
It is stated that when a C.Q. holds at a local optimum, then there exist Lagrange multipliers that satisfy KKT conditions.
But, I cannot get my head around this notion of local optimality. Does it mean locally optimal for the unconstrained problem? Does not local optimality imply the satisfaction of the KKT conditions?
It means locally optimal for the constrained problem.
If a constraint qualification does not hold, along with the required continuous differentiability of f(x) and g(x), a locally optimal solution need not satisfy the KKT conditions.