Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications

TL;DR

This paper establishes the equivalence between strong variational sufficiency and strong second order sufficient condition in nonlinear semidefinite programming, enabling convergence analysis of augmented Lagrangian and semi-smooth Newton methods without constraint qualifications.

Contribution

It proves the equivalence between strong variational sufficiency and strong SOSC for NLSDP without requiring constraint qualifications, advancing convergence analysis.

Findings

Equivalence between strong variational sufficiency and strong SOSC for NLSDP.

Convergence of ALM under strong SOSC without constraint qualifications.

Applicability of semi-smooth Newton method to ALM subproblems.

Abstract

Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for non-polyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP), without requiring the uniqueness of multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.