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We consider the convex optimization problem $\min \{f(x) : g_j(x)\leq 0,j=1,...,m\}$ where $f$ is convex, the feasible set K is convex and Slater'scondition holds, but the functions $g_j$ are not necessarily convex. We showthat for any representation of K that satisfies a mild nondegeneracyassumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and converselyevery KKT point is a minimizer. That is, the KKT optimality conditions arenecessary and sufficient as in convex programming where one assumes that the$g_j$ are convex. So in convex optimization, and as far as one is concernedwith KKT points, what really matters is the geometry of K and not so much itsrepresentation.

On representations of the feasible set in convex optimization