  Regularization plays a key role in a variety of optimization formulations of
inverse problems. A recurring theme in regularization approaches is the
selection of regularization parameters, and their 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. This paper characterizes the variational properties of the value
functions for a broad class of convex formulations, which are not all covered
by standard Lagrange multiplier theory. An inverse function theorem is given
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.
Numerical examples illustrate the theoretical results.
