Trade-off Invariance Principle for Minimizers of Regularized Functionals

In this paper, we consider functionals of the form H_α (u)=F(u)+α G(u) H α ( u ) = F ( u ) + α G ( u ) with α ∈ [0,+∞ ) α ∈ [ 0 , + ∞ ) , where u varies in a set U≠ ∅ U ≠ ∅ (without further structure). We first show that, excluding at most countably many values of α α , we have inf H_α G= H_α G inf H α ⋆ G = sup H α ⋆ G , where H_α ^ := min UH_α 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_α H α , which extends the above Principle to the case H_α ^ = ∅ H α ⋆ = ∅ . This fact implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of α 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.