  In this two-part study, we develop a general theory of the so-called exact
augmented Lagrangians for constrained optimization problems in Hilbert spaces.
In contrast to traditional nonsmooth exact penalty functions, these augmented
Lagrangians are continuously differentiable for smooth problems and do not
suffer from the Maratos effect, which makes them especially appealing for
applications in numerical optimization. Our aim is to present a detailed study
of various theoretical properties of exact augmented Lagrangians and discuss
several applications of these functions to constrained variational problems,
problems with PDE constraints, and optimal control problems.
  The first paper is devoted to a theoretical analysis of an exact augmented
Lagrangian for optimization problems in Hilbert spaces. We obtain several
useful estimates of this augmented Lagrangian and its gradient, and present
several types of sufficient conditions for KKT-points of a constrained problem
corresponding to locally/globally optimal solutions to be local/global
minimisers of the exact augmented Lagrangian.
