In this chapter we develop a novel approach for the solution of inequality constrained optimization problems. We first describe inexact Newton methods in Section 5.1 and investigate their local convergence in Section 5.2. In Section 5.3 we review strategies for the globalization of convergence and explain a different approach based on generalized level functions and monotonicity tests. An example in Section 5.4 illustrates the shortcomings of globalization strategies which are not based on the so called natural level function. We review the Restrictive Monotonicity Test (RMT) in Section 5.5 and propose a Natural Monotonicity Test (NMT) for Newton-type methods based on a Linear Iterative Splitting Approach (LISA). This combined approach allows for estimation of the critical constants which characterize convergence. We finally present how these results can be extended to global inexact SQP methods. We present efficient numerical solution techniques of the resulting sequence of Quadratic Programming Problems (QPs) in Chapters 8 and 9.