On the Role of the Mangasarian-Fromovitz Constraint Qualification for Penalty-, Exact Penalty- and Lagrange Multiplier Methods

In this paper we consider three embeddings (one-parametric optimization problems) motivated by penalty, exact penalty and Lagrange multiplier methods. We give an answer to the question under which conditions these methods are successful with an arbitrarily chosen starting point. Using the theory of one-parametric optimization (the local structure of the set of stationary points and of the set of generalized critical points, singularities, structural stability, pathfollowing and jumps) the so-called Mangasarian-Fromovitz condition and its extension play an important role. The analysis shows us that the class of optimization problems for which we can surely find a stationary point using a pathfollowing procedure for the modified penalty and exact penalty embedding is much larger than the class where the Lagrange multiplier embedding is successful. For the first class, the objective may be a “really non-convex” function, but for the second one we are restricted to convex optimization problems. This fact was a surprise at least for the authors