Stable Well-posedness and Tilt stability with respect to admissible functions

TL;DR

This paper explores the relationship between stable well-posedness and tilt stability of lower semicontinuous functions using admissible functions, establishing conditions under which these properties are equivalent or related.

Contribution

It introduces the concepts of $ ext{ extphi}$-SLWP and $ ext{ extpsi}$-TSLM, revealing their equivalence under a specific relationship between $ ext{ extphi}$ and $ ext{ extpsi}$, and links these to metric regularity conditions.

Findings

$ ext{ extphi}$-SLWP iff $ ext{ extpsi}$-TSLM when $ ext{ extpsi}=( ext{ extphi}')^{-1}$

Strong metric $ ext{ extphi}'$-regularity implies $ ext{ extphi}$-SLWP

Results generalize existing tilt stability findings for $ ext{ extphi}(t)=t^2$

Abstract

Note that the well-posedness of a proper lower semicontinuous function $f$ can be equivalently described using an admissible function. In the case when the objective function $f$ undergos the tilt perturbations in the sense of Poliquin and Rockafellar, adopting admissible functions $\varphi$ and $\psi$, this paper introduces and studies the stable well-posedness of $f$ with respect to $\varphi$ (in breif, $\varphi$-SLWP) and tilt-stable local minimum of $f$ with respect to $\psi$ (in brief, $\psi$-TSLM). In the special case when $\varphi(t)=t^2$ and $\psi(t)=t$, the corresponding $\varphi$-SLWP and $\psi$-TSLM reduce to the stable second local minimizer and tilt stable local minimum respectively, which have been extensively studied in recent years. We discover an interesting relationship between two admissible functions $\varphi$ and $\psi$: $\psi(t)=(\varphi')^{-1}(t)$, which implies that a proper lower semicontinous function $f$ on a Banach space has $\varphi$-SLWP if and only if $f$ has $\psi$-TSLM. Using the techniques of variational analysis and conjugate analysis, we also prove that the strong metric $\varphi'$-regularity of $\partial f$ is a sufficient condition for $f$ to have $\varphi$-SLWP and that the strong metric $\varphi'$-regularity of $\partial\overline{\rm co}(f+\delta_{B[\bar x,r]})$ for some $r>0$ is a necessary condition for $f$ to have $\varphi$-SLWP. In the special case when $\varphi(t)=t^2$, our results cover some existing main results on the tilt stability.