Strong duality and KKT conditions in nonconvex optimization with a single equality constraint and geometric constraint

Some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established. In particular, a hidden convexity of the conic hull of joint-range of the pair of functions associated to the original problem, is obtained. Applications to derive (a characterization of the validity of) KKT conditions without standard constraints qualification, are also discussed. It goes beyond the exact penalization technique. Several examples showing our results provide much more information than those appearing elsewhere, are given. Finally, the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail.