Can Slater's condition be used for non-convex objective functions?

Question

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?

Answer 1

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.