Critical Multipliers in Semidefinite Programming

TL;DR

This paper extends the concept of critical and noncritical multipliers from nonlinear programming to semidefinite programming, providing characterizations, conditions, and error bounds without additional assumptions.

Contribution

It introduces the notion of critical multipliers in SDP, characterizes them, and links noncriticality to error bounds under broad conditions, with explicit second-order optimality criteria.

Findings

Noncritical multipliers can be derived from error bounds in SDP.

Explicit second-order sufficient optimality conditions for noncriticality.

New error bounds for the primal variables involving perturbations and multipliers.

Abstract

It was proved in [14] that the existence of a noncritical multiplier for a (smooth) nonlinear programming problem is equivalent to an error bound condition for the Karush-Kuhn-Thcker (KKT) system without any assumptions. This paper investigates whether this result still holds true for a (smooth) nonlinear semidefinite programming (SDP) problem. We first introduce the notion of critical and noncritical multipliers for a SDP problem and obtain their complete characterizations in terms of the problem data. We prove for the SDP problem, the noncriticality property can be derived from the error bound condition for the KKT system without any assumptions, and this fact is revealed by some simple examples. Besides we give an appropriate second-order sufficient optimality condition characterizing noncriticality explicitly. We propose a set of assumptions from which the error bound condition for the KKT system can be derived from the noncriticality property. Finally we establish a new error bound for x-part, which is expressed by both perturbation and the multiplier estimation.