The class of inverse problems for a nonlinear elliptic variational inequality is considered. The nonlinear elliptic operator is assumed to be a monotone potential. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients which is compact in H-1(0, xi*). It is shown that the nonlinear operator is pseudomonotone for the given class of coefficients. For the corresponding direct problem H-1- coefficient convergence is proved. Based on this result the existence of a quasisolution of the inverse problem is obtained. As an important application an inverse diagnostic problem for an axially symmetric elasto-plastic body is considered. For this problem the numerical method and computational results are also presented.