Trade-off Invariance Principle for Minimizers of Regularized Functionals

Abstract In this paper, we consider functionals of the form $$H_\alpha (u)=F(u)+\alpha G(u)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:mi>G</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with $$\alpha \in [0,+\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where u varies in a set $$U\ne \emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> (without further structure). We first show that, excluding at most countably many values of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , we have $$\inf _{H_\alpha ^\star }G= \sup _{H_\alpha ^\star }G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>inf</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> <mml:mo>⋆</mml:mo> </mml:msubsup> </mml:msub> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>sup</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> <mml:mo>⋆</mml:mo> </mml:msubsup> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , where $$H_\alpha ^\star {:}{=}\arg \min _UH_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> <mml:mo>⋆</mml:mo> </mml:msubsup> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>arg</mml:mo> <mml:msub> <mml:mo>min</mml:mo> <mml:mi>U</mml:mi> </mml:msub> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> </mml:math> , 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 $$H_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> , which extends the above Principle to the case $$H_\alpha ^\star = \emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> <mml:mo>⋆</mml:mo> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> . This fact implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , 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.