Critical Multipliers in Semidefinite Programming

It was proved in Izmailov and Solodov (2014). Newton-Type Methods for Optimization and Variational Problems, Springer] 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. The answer is negative: the existence of noncritical multiplier does not imply the error bound condition for the KKT system without additional conditions, which is illustrated by an example. In this paper, we obtain characterizations, in terms of the problem data, the critical and noncritical multipliers for a SDP problem. We prove that, for the SDP problem, the noncriticality property can be derived from the error bound condition for the KKT system without any assumptions, and we give an example to show that the noncriticality does not imply the error bound for the KKT system. We propose a set of assumptions under which the error bound condition for the KKT system can be derived from the noncriticality property.a Finally, we establish a new error bound for x-part, which is expressed by both perturbation and the multiplier estimation.