The consideration of constraint conditions plays an important role in many areas of continuum and structural mechanics. For example, in the modeling of shear force hinges in frame structures, in the modeling of rigid inclusions in a body originally modeled as deformable, or in the modeling of deformation-dependent Dirichlet boundary conditions. In the context of the finite element method, the constraints refer to the nodal degrees of freedom. If several nodes are involved, they are referred to as multi-point constraints. These must be formulated non-linearly if the degrees of freedom involved change significantly during the simulation (e.g. as a result of a heavily deformed FE mesh). One method for considering constraints is master-slave elimination, which, in contrast to Lagrange multipliers and the penalty method, offers the advantage of reducing the dimension of the problem. However, the existing master-slave elimination method is limited to linear constraints. The presentation introduces a new master-slave elimination method for the treatment of nonlinear multi-point constraints. The method is based on a mathematically rigorous derivation, where an 'optimization problem with constraints' is chosen as the starting point. It is transformed into a 'modified optimization problem without constraints' using the implicit function theorem. In order to perform this transformation, the set of slave degrees of freedom must be chosen in such a way that the Jacobian matrix derived from the constraints fulfills several conditions. The derivation is general and is not restricted to specific constraints. As part of the algorithmic implementation, it is possible to separate the method itself from the specification of the constraint. In the context of several numerical examples, the new method is compared with the existing methods. The results show that the new method is as accurate, robust and flexible as the Lagrange multipliers and is more efficient by reducing the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.