Perturbation analysis of a class of composite optimization problems

Abstract In this paper, we conduct a perturbation analysis of a class of composite optimization problems, providing a unified framework for addressing both theoretical and algorithmic aspects of constrained optimization problems. By exploiting the properties of the graphical derivative, limiting/Mordukhovich coderivative, and regular coderivative, we establish a strong second‐order sufficient condition (SSOSC) for the composite optimization problem. This condition represents a novel contribution that cannot be directly inferred from existing definitions tailored to specific problems. Under certain mild assumptions on the objective function, it is demonstrated that SSOSC together with the nondegeneracy condition, the nonsingularity of Clarke's generalized Jacobian of the nonsmooth system at a Karush–Kuhn–Tucker (KKT) point, and strong regularity of the KKT point are equivalent. These findings provide a robust analytical tool for evaluating the stability of the KKT point.