Identification problems for variational inequalities frequently occur in practical applications. For instance, the estimation of the height function *u* between two rotating surfaces in elastohydrodynamic lubrication problems belongs to the present problem class. In this situatuation the variational inequality stems from the Reynolds lubrication equation and the additional requirement that the pressure *y* is nonnegative.

A common technique to identify the parameter *u* from measurements *z *of *y *is based on a regularized least squares formulation

Note that the (model) variational inequality is the necessary and sufficient first order condition for the lower level minimization problem in the above least squares problem. The characterization of an optimal solution (*y*,u**) of the above *bilevel* optimization problem is usually based on multipliers. If we replace the *lower level *minimization problem by its optimality system, i.e. the equilibrium system from above, as a so called *equilibrium constraint*, then existence of multipliers may fail even in practically relevant situations.

?Typical results on existence of multipliers available from the literature either guarantee multipliers in very general situations but are not amenable for numerical realization, or the corresponding first order conditions are numerically implementable but existence of multipliers fails in specific important cases.

The numerical realization is based on a finite element discretization of the above first order necessary conditions. The pointwise conditions are enforced at the nodes of the discretization. The resulting nonlinear system is solved by a stabilized *Gauss-Newton* method combined with an interior technique for *u*. The latter approach allows to rewrite the variational inequality in the first order system as an equality. The identification is done on a suitable observation set (see the figure below). The graphs below show the observation part of the domain and the respective reconstructions.