Trade-off Invariance Principle for minimizers of regularized functionals

TL;DR

This work introduces a Trade-off Invariance Principle for minimizers of regularized functionals of the form $H_\alpha(u)=F(u)+\alpha G(u)$, showing that, for all but countably many $\alpha$, the limiting value of $G$ along minimizers is invariant, and this extends to minimizing sequences even when minimizers do not exist. It further extends the principle to multi-regularization, critical points under differentiable structure and Łojasiewicz assumptions, and derives measurable, almost-everywhere statements about the limiting behavior of $G_j$ across multiple regularizers. The results yield practical consequences: (i) differentiability of the value function characterizes points of invariance; (ii) penalty formulations for equality constraints inherit almost-everywhere invariance; (iii) in Sobolev and BV spaces, minimizing sequences exhibit almost-everywhere strong convergence under weak-to-strong regularization; (iv) under Gamma-convergence, quasi-minimizers remain strongly pre-compact and converge to minimizers of the limit problem. Collectively, the theory provides a robust framework linking trade-offs among regularization terms to convergence properties, variational limits, and optimization under nonconvexity.

Abstract

In this paper, we consider functionals of the form $H_α(u)=F(u)+αG(u)$ with $α\in[0,+\infty)$, where $u$ varies in a set $U\neq\emptyset$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $α$, we have $\inf_{H_α^\star}G= \sup_{H_α^\star}G$, where $H_α^\star := \arg\min_UH_α$, 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 $H_α$, which extends the above Principle to the case $H_α^\star= \emptyset$. 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 $α$, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent.

Theorems & Definitions (44)

• Theorem 1.1: Trade-off Invariance Principle I
• Remark 1
• Theorem 1.2: Trade-off Invariance Principle II
• Definition 1
• Corollary 1.3
• proof
• Remark 2
• Remark 3
• Remark 4
• Remark 5