Stability properties of the Tikhonov regularization for nonmonotone inclusions

Abstract

We study the Tikhonov regularization for perturbed inclusions of the form T (x) y * where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties and y * is an element near 0. We investigate the case when T is metrically regular and strongly regular and we show the existence of both a solution x * to the perturbed inclusion and a Tikhonov sequence which converges to x * . Finally, we show that the Tikhonov sequences associated to the perturbed problem inherit the regularity properties of the inverse of T .

A Lyusternik–Graves theorem for the proximal point method

“…We also present another result, Theorem 3.2, regarding the existence of a (super)linearly convergent proximal point sequence under metric regularity of T , which not only improves both [1, Theorem 3.1] and [2, Theorem 3.2], but is also essential for proving the main theorems. Finally, in the same way as in [10,11], it may be interesting to note that such results show in particular (in a certain perspective), that the proximal method is really wellposed since actually the whole proximal iterates of the problem depend continuously on the data of the problem. The results presented are theoretical; we are interested in studying how the regularity properties (and moduli) carry away from the mapping T to the set of convergent proximal point sequences, and vice versa.…”

Regularisation, optimisation, subregularity

“…We now briefly discuss how we can use some popular nonsmooth optimisation methods to construct x δ satisfying the accuracy estimate [13] and the parameter convergence conditions [19]. We start with forward-backward splitting, mainly applicable to the 1 -regularised regression of theorem 4.5, in which case it is also known as iterative soft-thresholding [30][31][32].…”

“…Proof. We use corollary 3.7, for which we need to verify [19] for some e δ satisfying [13]. Assumption 3.5 and [12] of assumption 3.1 we have assumed.…”

“…after taking N δ iterations from the fixed initial iterate x 0 . Thus the condition [19] of corollary 3.7 is satisfied by choosing α δ > 0 and N δ such that (49) holds.…”

“…Finally, metric regularity-a concept stronger than metric subregularity and distinct from strong metric subregularity-has been used in [19] to study the regularisation of set-valued inclusions 0 ∈ T(x) via the Tikhonov-style expression 0 ∈ [T + α δ Id](x δ ). The incorporation of the identity map Id requires T to be a set-valued map form X to X, whereas subdifferentials are set-valued maps from X to X * .…”