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As wee known, in convex optimization problem, we get strong duality if Slater's condition holds. I often use Slater condition to indicate whether an optimal solution of primary problem satisfies KKT system.

Can I use Slater's condition when the constrained inequality functions are convex, constrained equality functions are affine but the objective function is not convex?

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  • $\begingroup$ Yes, Slater condition for convex constraints implies that KKT system is satisfied at locally optimal points. Unfortunately, I do not have a reference. $\endgroup$
    – daw
    Aug 6, 2021 at 8:20
  • $\begingroup$ You got a quite clear answer on your Reddit post. $\endgroup$
    – KBS
    Feb 12, 2022 at 17:13

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See proposition 3.3.9 in the book Nonlinear Programming (second edition) by Dimitri Bertsekas. You will need continuous differentiability of the objective function and inequality constraints. The proof is an application of the Mangasarian-Fromovitz constraint qualification.

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