Algorithms based on polyhedral outer approximations provide a powerful approach to solving mixed-integer nonlinear optimization problems. An initial relaxation of the feasible set is strengthened by iteratively adding linear inequalities and separating infeasible points. However, when the constraints are nonconvex, computing such separating hyperplanes becomes challenging. In this article, the moment-/sums-of-squares hierarchy is used in the case that the objective and the constraints are polynomial. Although the calculated hyperplanes are not optimal in general, convergence to an optimal separating hyperplane can be shown under some general conditions. Then, this idea is embedded in an algorithmic framework for mixed-integer nonlinear problems. The algorithm is modified with two heuristic approaches inspired by sparsity structures to tackle instances with a high number of variables and constraints. A numerical study on a large set of problems from the MINLPLib demonstrates the potential of our method.