Trade-off Invariance Principle for minimizers of regularized functionals

Abstract

In this paper, we consider functionals of the form $H_{\hat{I}}\pm (u)=F(u)+\hat{I}\pm G(u)$ with $\hat{I}\pm \in [0,+\infty )$, where $u$ varies in a set $U\ne \varnothing$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $\hat{I}\pm$, we have ${\inf }_{H_{\hat{I}}{\pm }^{\star }}G={\sup }_{H_{\hat{I}}{\pm }^{\star }}G$, where $H_{\hat{I}}{\pm }^{\star }:=\arg {\min }_U{H}_{\hat{I}}\pm$, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional $G$ along minimizing sequences for $H_{\hat{I}}\pm$, which extends the above Principle to the case $H_{\hat{I}}{\pm }^{\star }=\varnothing$. Moreover, we show to what extent these findings generalize to multi-regularized functionals and -- in the presence of an underlying differentiable structure -- to critical points. Finally, the main result implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $\hat{I}\pm$, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent.

20 pages, extension to multi-regularization and to critical points, differentiability of the value function