The aim of the talk is to present some recent improvements on the theory of elastic-plastic torsion problem, specially characterizing the Lagrange Multiplier associated to the problem and showing the relationship with the obstacle problem. The existence of the Lagrange multiplier for the elastic-plastic torsion problem will be shown in more general settings with respect to the results in literature, and considering nonlinear monotone operators. Moreover these questions will be studied for variational problems with nonconstant gradient constraints. The main tool for the study is a recently developed strong duality theory, based on a constraint qualification assumption. This theory holds in infinite-dimensional settings and allows to apply the Lagrangian multipliers method to infinite-dimensional problems.