Stratification for Nonlinear Semidefinite Programming
Chenglong Bao, Chao Ding, Fuxiaoyue Feng, Jingyu Li
TL;DR
The paper introduces a stratification framework for NLSDP by decomposing $\mathbb{S}^n$ into inertia-based strata and lifting this structure to the primal–dual space, where the KKT mapping becomes smooth on each stratum. It defines stratum-restricted regularity and connects it to weak second-order and Robinson-type conditions via transversality, showing these weak conditions govern local behavior on each stratum while classical strong regularity arises from uniform validity across nearby strata. A global Stratified Gauss--Newton method with correction (SGN) is proposed to solve the KKT system by exploiting stratified geometry, ensuring global convergence to directional stationary points and local quadratic convergence to KKT pairs with active-stratum identification under W-SOC and SRCQ. The framework links perturbation analysis, variational geometry, and algorithm design, offering a principled path to relax classical regularity assumptions in NLSDP and related nonpolyhedral problems while retaining fast local convergence on the active stratum.
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.