In this paper, we consider functionals of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H_\alpha (u)=F(u)+\alpha G(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>α</mi> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \in [0,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where u varies in a set <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$U\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation> (without further structure). We first show that, excluding at most countably many values of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, we have <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\inf _{H_\alpha ^\star }G= \sup _{H_\alpha ^\star }G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">inf</mo> <msubsup> <mi>H</mi> <mi>α</mi> <mo>⋆</mo> </msubsup> </msub> <mi>G</mi> <mo>=</mo> <msub> <mo movablelimits="true">sup</mo> <msubsup> <mi>H</mi> <mi>α</mi> <mo>⋆</mo> </msubsup> </msub> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H_\alpha ^\star {:}{=}\arg \min _UH_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>α</mi> <mo>⋆</mo> </msubsup> <mo>:</mo> <mo>=</mo> <mo>arg</mo> <msub> <mo movablelimits="true">min</mo> <mi>U</mi> </msub> <msub> <mi>H</mi> <mi>α</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional G along minimizing sequences for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H_\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>, which extends the above Principle to the case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$H_\alpha ^\star = \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>α</mi> <mo>⋆</mo> </msubsup> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation>. This fact implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, 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.

Trade-off Invariance Principle for Minimizers of Regularized Functionals

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2026年1月12日

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Massimo Fornasier,
Jona Klemenc,
Alessandro Scagliotti

摘要

In this paper, we consider functionals of the form $H_\alpha (u)=F(u)+\alpha G(u)$ $H_{α} (u) = F (u) + α G (u)$ with $\alpha \in [0,+\infty )$ $α \in [0, + \infty)$ , where u varies in a set $U\ne \emptyset $ $U \neq \emptyset$ (without further structure). We first show that, excluding at most countably many values of $\alpha $ $α$ , we have $\inf _{H_\alpha ^\star }G= \sup _{H_\alpha ^\star }G$ ${inf}_{H_{α}^{⋆}} G = {sup}_{H_{α}^{⋆}} G$ , where $H_\alpha ^\star {:}{=}\arg \min _UH_\alpha $ $H_{α}^{⋆} : = arg {min}_{U} H_{α}$ , 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 $ $H_{α}$ , which extends the above Principle to the case $H_\alpha ^\star = \emptyset $ $H_{α}^{⋆} = \emptyset$ . This fact implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $\alpha $ $α$ , 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.