Strong calmness of perturbed KKT system for a class of conic programming with degenerate solutions

TL;DR

This paper investigates the strong calmness property of the KKT solution mapping in perturbed conic programming, establishing equivalences with error bounds and providing conditions for noncritical multipliers, extending prior results.

Contribution

It introduces new conditions for strong calmness in conic programming, linking it to error bounds and multiplier properties, extending previous nonlinear and semidefinite programming results.

Findings

Strong calmness is equivalent to a local error bound.

Pseudo-isolated calmness of stationary points relates to strong calmness.

Conditions involving noncritical multipliers ensure strong calmness.

Abstract

This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for solutions of perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set map at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in \cite{Hager99,Izmailov12} for nonlinear programming and in \cite{Cui16,Zhang17} for semidefinite programming.