Tilt stability of a class of nonlinear semidefinite programs

TL;DR

This paper investigates the tilt stability of local solutions in nonlinear semidefinite programs with convex constraints, providing new point-based characterizations under various conditions, including cases with positive semidefinite Hessians.

Contribution

It introduces the first point-based necessary and sufficient conditions for tilt stability in nonlinear semidefinite programs without relying on constraint nondegeneracy.

Findings

Derived two point-based sufficient conditions for tilt stability.

Established a necessary characterization with a gap from sufficiency for certain cases.

Provided a weaker necessary and sufficient characterization under specific multiplier restrictions.

Abstract

This paper concerns the tilt stability of local optimal solutions to a class of nonlinear semidefinite programs, which involves a twice continuously differentiable objective function and a convex feasible set. By leveraging the second subderivative of the extended-valued objective function and imposing a suitable restriction on the multiplier, we derive two point-based sufficient characterizations for tilt stability of local optimal solutions around which the objective function has positive semidefinite Hessians, and for a class of linear positive semidefinite cone constraint set, establish a point-based necessary characterization with a certain gap from the sufficient one. For this class of linear positive semidefinite cone constraint case, under a suitable restriction on the set of multipliers, we also establish a point-based sufficient and necessary characterization, which is weaker than the dual constraint nondegeneracy when the set of multipliers is singleton. As far as we know, this is the first work to study point-based sufficient and/or necessary characterizations for nonlinear semidefinite programs with convex constraint sets without constraint nondegeneracy conditions.