Characterizations of the Aubin property of the KKT-mapping in composite optimization by SC derivatives and quadratic bundles
Helmut Gfrerer, Jiri V. Outrata
TL;DR
This work addresses the stability of the KKT mapping $S_{KKT}$ in composite optimization problems of the form $φ(x)=f(x)+g(F(x))$. It replaces the traditional limiting coderivative and strict graphical derivative approach with more tractable tools: SC derivatives and Rockafellar's quadratic bundles, enabling practical characterizations of the Aubin property and related primal–dual stability. The authors prove a network of equivalent conditions linking the Aubin property, the second-order qualification condition (SOQC), strong variational sufficiency, and tilt-stability, under natural regularity and chain-rule assumptions, with exact chain rules for the graphical derivatives of $∂(g∘F_b)$ and $∂(g∘F)$. These results yield computable criteria for verifying Lipschitzian localizations of the KKT system and provide insights into the sensitivity of solutions to perturbations, which is valuable for algorithmic implementations and robust optimization.
Abstract
For general set-valued mappings, the Aubin property is ultimately tied to limiting coderivatives by the Mordukhovich criterion. Likewise, the existence of single-valued Lipschitzian localizations is related to strict graphical derivatives. In this paper we will show that for the special case of the KKT-mapping from composite optimization, the Aubin property and the existence of single-valued Lipschitzian localizations can be characterized by SC derivatives and quadratic bundles, respectively, which are easier accessible than limiting coderivatives and strict graphical derivatives.