  In this paper, we consider a finite dimensional optimization problem
minimizing a continuous objective on a compact domain subject to a
multi-dimensional constraint function. For the latter, we only assume the
availability of a Lipschitz property. In recent literature methods based on
non-convex outer approximation are proposed for tackling one dimensional
equality constraints on bounded polyhedral domains, which are Lipschitz with
respect to the maximum norm. To the best of our knowledge, however, there
exists no non-convex outer approximation method for a general problem class. We
introduce a meta-level solution framework to solve such problems and tackle the
underlying theoretical foundations. Considering the feasible domain without the
constraint function as manageable, our method relaxes the multidimensional
constraint and iteratively refines the feasible region by means of norm-induced
cuts, relying on an oracle for the resulting sub-problems. We show the method's
correctness and investigate the problem complexity. In order to account for
discussions about functionality, limits, and extensions, we present
computational examples including illustrations.
