Stratification for Nonlinear Semidefinite Programming

TL;DR

This paper develops a stratification framework for nonlinear semidefinite programming that enhances understanding of the geometry of the KKT system and proposes a globally convergent stratified Gauss-Newton algorithm.

Contribution

It introduces a novel stratification approach for NLSDP, linking geometric properties with regularity conditions and designing a new algorithm with proven convergence properties.

Findings

The stratified Gauss-Newton method converges globally to stationary points.

Under certain conditions, the method achieves local quadratic convergence.

The framework clarifies the geometric interpretation of regularity conditions.

Abstract

This paper introduces a stratification framework for nonlinear semidefinite programming (NLSDP) that reveals and utilizes the geometry behind the nonsmooth KKT system. Based on the \emph{index stratification} of $\mathbb{S}^n$ and its lift to the primal--dual space, a stratified variational analysis is developed. Specifically, we define the stratum-restricted regularity property, characterize it by the verifiable weak second order condition (W-SOC) and weak strict Robinson constraint qualification (W-SRCQ), and interpret the W-SRCQ geometrically via transversality, which provides its genericity over ambient space and stability along strata. The interactions of these properties across neighboring strata are further examined, leading to the conclusion that classical strong-form regularity conditions correspond to the local uniform validity of stratum-restricted counterparts. On the algorithmic side, a stratified Gauss--Newton method with normal steps and a correction mechanism is proposed for globally solving the KKT equation through a least-squares merit function. We demonstrate that the algorithm converges globally to directional stationary points. Moreover, under the W-SOC and the strict Robinson constraint qualification (SRCQ), it achieves local quadratic convergence to KKT pairs and eventually identifies the active stratum.