# Characterization of tilt stability via subgradient graphical derivative with applications to nonlinear programming

## TL;DR

This paper introduces a new way to characterize tilt stability in finite-dimensional optimization problems using the subgradient graphical derivative, providing second-order conditions and applications to nonlinear programming.

## Contribution

It offers a novel characterization of tilt stability via the subgradient graphical derivative and applies it to nonlinear programming under metric subregularity.

## Key findings

- Characterization of tilt-stable local minimizers using subgradient graphical derivative.
- Second-order conditions for tilt stability in nonlinear programming.
- Stationary points satisfying strong second-order conditions are tilt-stable.

## Abstract

This paper is devoted to the study of tilt stability in finite dimensional optimization via the approach of using the subgradient graphical derivative. We establish a new characterization of tilt-stable local minimizers for a broad class of unconstrained optimization problems in terms of a uniform positive definiteness of the subgradient graphical derivative of the objective function around the point in question. By applying this result to nonlinear programming under the metric subregularity constraint qualification, we derive a second-order characterization and several new sufficient conditions for tilt stability. In particular, we show that each stationary point of a nonlinear programming problem satisfying the metric subregularity constraint qualification is a tilt-stable local minimizer if the classical strong second-order sufficient condition holds.