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Trade-Off Invariance Principle for Minimizers of Regularized Functionals

MCML Authors

Massimo Fornasier

Prof. Dr.

Principal Investigator

Applied Numerical Analysis

Jona Klemenc

→ Group Massimo Fornasier
Applied Numerical Analysis

Alessandro Scagliotti

→ Group Massimo Fornasier
Applied Numerical Analysis

Abstract

In this paper, we consider functionals of the form Hα(u)=F(u)+αG(u) with α∈[0,+∞), where u varies in a set U≠∅ (without further structure). We first show that, excluding at most countably many values of α, we have that infH⋆αG=supH⋆αG, where H⋆α:=argminUHα, which is assumed to be non-empty. We further prove a stronger result that concerns the {invariance of the} limiting value of the functional G along minimizing sequences for Hα. This fact in turn implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of α, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent. Finally, we show to what extent these findings generalize to multi-regularized functionals and—in the presence of an underlying differentiable structure—to critical points.

article FKS26