Recently, the Hypervolume Newton method (HVN) has been proposed as fast and precise indicator-based method for solving unconstrained bi-objective optimization problems with objective functions that are at least twice continuously differentiable. The HVN is defined on the space of (vectorized) fixed cardinality sets of decision space vectors for a given multi-objective optimization problem (MOP) and seeks to maximize the hypervolume indicator adopting the Newton-Raphson method for deterministic numerical optimization. To extend its scope to non-convex optimization problems the HVN method was hybridized with a multi-objective evolutionary algorithm (MOEA), which resulted in a competitive solver for continuous unconstrained bi-objective optimization problems. In this paper, we extend the HVN to constrained MOPs with in principle any number of objectives. We demonstrate the applicability of the extended HVN on a set of challenging benchmark problems and show that the new method can be readily be applied to solve equality constraints with a high precision problems, and to some extend also inequalities. We finally use HVN as local search engine within a MOEA and show the benefit of this hybrid method on several benchmark problems.