Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring question in regularization approaches is the selection of regularization parameters, and its effect on the solution and on the optimal value of the optimization problem. The sensitivity of the value function to the regularization parameter can be linked directly to the Lagrange multipliers. In this paper, we fully characterize the variational properties of the value functions for a broad class of convex formulations, which are not all covered by standard Lagrange multiplier theory. We also present an inverse function theorem that links the value functions of different regularization formulations (not necessarily convex). These results have implications for the selection of regularization parameters, and the development of specialized algorithms. We give numerical examples that illustrate the theoretical results.