Local Convexification of the Lagrangian Function in Nonconvex Optimization

It is well-known that a basic requirement for the development of local duality theory in nonconvex optimization is the local convexity of the Lagrangian function. This paper shows how to locally convexify the Lagrangian function and thus expand the class of optimization problems to which dual methods can be applied. Specifically, we prove that, under mild assumptions, the Hessian of the Lagrangian in some transformed equivalent problem formulations becomes positive definite in a neighborhood of a local optimal point of the original problem.