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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?

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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.

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  • $\begingroup$ can you provide an example where the local optimality of a constrained problem does not imply satisfaction of KKT conditions? $\endgroup$
    – shnnnms
    Apr 2, 2022 at 19:30
  • $\begingroup$ The simplest possible example is by @daw at math.stackexchange.com/questions/2513300/… $\endgroup$ Apr 2, 2022 at 21:36
  • $\begingroup$ is there any other example with a feasible set that is not of Lebesgue measure zero? $\endgroup$
    – shnnnms
    Apr 2, 2022 at 23:56
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    $\begingroup$ Yes. Example 4.5.3 and Solution 4.5.4 of math.uni-bielefeld.de/~drust/opt2017-part4.pdf provides such an example. $\endgroup$ Apr 3, 2022 at 16:15

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