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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?

shnnnms
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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.

Mark L. Stone
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  • can you provide an example where the local optimality of a constrained problem does not imply satisfaction of KKT conditions? – shnnnms Apr 02 '22 at 19:30
  • The simplest possible example is by @daw at https://math.stackexchange.com/questions/2513300/is-kkt-conditions-necessary-and-sufficient-for-any-convex-problems/2513724#2513724 – Mark L. Stone Apr 02 '22 at 21:36
  • is there any other example with a feasible set that is not of Lebesgue measure zero? – shnnnms Apr 02 '22 at 23:56
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    Yes. Example 4.5.3 and Solution 4.5.4 of https://www.math.uni-bielefeld.de/~drust/opt2017-part4.pdf provides such an example. – Mark L. Stone Apr 03 '22 at 16:15