Trade-off Invariance Principle for Minimizers of Regularized Functionals

Abstract In this paper, we consider functionals of the form H_ (u) =F (u) + G (u) H α (u) = F (u) + α G (u) with [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, we have ₇_ ^ G= ₇_ ^ G inf H α ⋆ G = sup H α ⋆ G, where H_ ^: = UH_ 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_ H α, which extends the above Principle to the case H_ ^ = H α ⋆ = ∅. This fact 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.