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

On Lagrange multipliers of the KKT system in Hilbert spaces