A sufficient and necessary condition for nonconvex constrained optimization

The conventional Lagrangian approach to solving constrained optimization problems leads to optimally conditions which are either necessary, or sufficient, but not both unless the underlying cost and constraint functions are also convex. We introduce a new approach based on the Tchebyshev norm. This leads to an optimality condition which is both sufficient and necessary, without any convexity assumption. This optimality condition can be used to devise a conceptually simple method for solving nonconvex inequality constrained optimization problems.