Global Lagrange multiplier rule and smooth exact penalty functions for equality constraints

Abstract

Smooth exact penalty functions based on Courant’s and Fletcher’s ideas are reconsidered. After a short survey, the original ideas are combined with the global Lagrange multiplier rule formulated by the first and second covariant derivatives of the objective function with respect to the induced Riemannian metric of the constraint manifold. The tensor approach is described by the usual tools of nonlinear optimization giving a clearer geometric background of these methods.