# Stratification for Nonlinear Semidefinite Programming

arXiv: https://arxiv.org/abs/2601.08362 · HTML: https://mlsanity.com/paper/2601.08362/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jan 2026.

## Built on by (most cited first, 1 of 1)

- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

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 index stratification of mathbbSn 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, with 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 second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) at an accumulation point, with a suitable correction threshold, the whole sequence converges superlinearly to this point, which is a KKT pair, and eventually identifies the active stratum. The rate is quadratic if the problem data are additionally of class LC2 near the solution.