  In this paper we develop a new decomposition framework to deal with Lagrange
multipliers of the Karush-Kuhn-Tucker (KKT) system of constrained optimization
problems and variational inequalities in Hilbert spaces. It is different from
existing frameworks based on separation theorems. We introduce the essential
Lagrange multiplier and establish the basic theory of this new multiplier. The
essential Lagrange multiplier poses essentially different existence results in
finite and infinite-dimensional spaces. It can also be used to give an
essential characterization of the convergence of multipliers generated by the
classical augmented Lagrangian method. Our analysis reveals that the essential
Lagrange multiplier is at the core of both theories and applications of
Lagrange multipliers.
