  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.
