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Regularization plays a key role in a variety of optimization formulations ofinverse problems. A recurring theme in regularization approaches is theselection of regularization parameters, and their effect on the solution and onthe optimal value of the optimization problem. The sensitivity of the valuefunction to the regularization parameter can be linked directly to the Lagrangemultipliers. This paper characterizes the variational properties of the valuefunctions for a broad class of convex formulations, which are not all coveredby standard Lagrange multiplier theory. An inverse function theorem is giventhat links the value functions of different regularization formulations (notnecessarily convex). These results have implications for the selection ofregularization parameters, and the development of specialized algorithms.Numerical examples illustrate the theoretical results.

Variational properties of value functions