Second-Order Characterizations of Tilt Stability in Composite Optimization

TL;DR

This paper provides a comprehensive second-order theoretical framework for understanding tilt stability in composite optimization problems, enhancing the foundation for stability analysis and numerical methods.

Contribution

It introduces unified second-order characterizations of tilt stability under weak conditions for a broad class of composite optimization problems.

Findings

Unified neighborhood and point-based characterizations of tilt stability.

Applicability under the weakest metric subregularity constraint qualification.

Foundation for future variational stability and algorithm development.

Abstract

Tilt stability is a fundamental concept of variational analysis and optimization that plays a pivotal role in both theoretical issues and numerical computations. This paper investigates tilt stability of local minimizers for a general class of composite optimization problems in finite dimensions, where extended-real-valued objectives are compositions of parabolically regular and smooth functions. Under the weakest metric subregularity constraint qualification and other verifiable conditions, we establish unified neighborhood and pointbased characterizations of tilt stability via second-order generalized differentiation. The obtained results provide a rigorous theoretical foundation for further developments on variational stability and numerical algorithms of optimization and related topics.