  In this paper, we study the perturbation analysis of a class of composite
optimization problems, which is a very convenient and unified framework for
developing both theoretical and algorithmic issues of constrained optimization
problems. The underlying theme of this paper is very important in both
theoretical and computational study of optimization problems. Under some mild
assumptions on the objective function, we provide a definition of a strong
second order sufficient condition (SSOSC) for the composite optimization
problem and also prove that the following conditions are equivalent to each
other: the SSOSC and the nondegeneracy condition, the nonsingularity of
Clarke's generalized Jacobian of the nonsmooth system at a Karush-Kuhn-Tucker
(KKT) point, and the strong regularity of the KKT point. These results provide
an important way to characterize the stability of the KKT point.
  As for the convex composite optimization problem, which is a special case of
the general problem, we establish the equivalence between the primal/dual
second order sufficient condition and the dual/primal strict Robinson
constraint qualification, the equivalence between the primal/dual SSOSC and the
dual/primal nondegeneracy condition. Moreover, we prove that the dual
nondegeneracy condition and the nonsingularity of Clarke's generalized Jacobian
of the subproblem corresponding to the augmented Lagrangian method are also
equivalent to each other. These theoretical results lay solid foundation for
designing an efficient algorithm.
