Complete Characterizations of Well-Posedness in Parametric Composite Optimization

TL;DR

This paper provides a comprehensive characterization of well-posedness for the KKT system in parametric composite optimization, introducing new second-order conditions and establishing equivalences among stability, regularity, and nonsingularity properties.

Contribution

It introduces a novel second-order variational function, extends classical second-order conditions, and establishes equivalences among various stability and regularity conditions in composite optimization.

Findings

Characterizes well-posedness of KKT systems in composite optimization.

Introduces a new second-order variational function governing second-order behavior.

Proves equivalence between Lipschitz-like property and nonsingularity of the generalized Jacobian.

Abstract

This paper characterizes the well-posedness of Karush-Kuhn-Tucker system for perturbed composite optimization. Using the parabolic regularity, we introduce a novel second-order variational function, shown to be the pivotal object governing second-order behavior. This foundational result yields the strong second-order sufficient condition introduced here for the general class of composite optimization problems to naturally extend the classical second-order sufficient condition in nonlinear programming. Then we obtain several equivalent characterizations of the second-order qualification condition (SOQC) and highlight its equivalence to the constraint nondegeneracy condition under the $\mathcal{C}^{2}$-cone reducibility assumption. These insights lead us to multiple equivalent conditions for the major Lipschitz-like/Aubin property of KKT systems, including SOQC combined with the new second-order subdifferential condition and SOQC combined with tilt stability of local minimizers. Under the $\mathcal{C}^{2}$-cone reducibility, we settle the long-standing question by proving the equivalence between the Aubin property of KKT systems around local minimizers and the classical notion of strong regularity under some additional assumptions. Finally, we demonstrate that the Lipschitz-like property is equivalent to the nonsingularity of the generalized Jacobian associated with the KKT system under a certain verifiable assumption. These results provide a unified and rigorous framework for analyzing stability and sensitivity of solutions to composite optimization problems, as well as for the design and justification of numerical algorithms.