In this paper we analyze two regularization methods for nonlinear ill-posed problems. The first is a penalty method called Tikhonov regularization, in which one solves an unconstrained optimization problem while the second is based on a constrained optimization problem. For each method we examine the well- posedness of the respective optimization problem. We then show strong convergence of the regularized 'solutions' to the true solution. (Note that this is well known for the application of these methods to linear problems.) In this analysis we consider such factors as the convergence of perturbed data to the true data, inexact solution of the respective optimization problems, and the choice of the regularization parameters.