Complete Characterizations of Well-Posedness in Parametric Composite Optimization

Abstract: This paper provides complete characterization of well-posedness for Karush-Kuhn-Tucker (KKT) systems associated with general problems of perturbed composite optimization. Leveraging the property of parabolic regularity for composite models, we show that the second-order subderivative of the cost function reduces to the novel second-order variational function playing a crucial role in the subsequent analysis. This foundational result implies that the strong second-order sufficient condition (SSOSC) introduced in this work for the general class of composite optimization problems naturally extends the classical second-order sufficient condition in nonlinear programming. Then we obtain several equivalent characterizations of the second-order qualification condition (SOQC) and highlight its equivalence to the constraint nondegeneracy condition under the $\mathcal{C}^{2}$-cone reducibility assumption. These insights lead us to multiple equivalent conditions for the major Lipschitz-like/Aubin property of KKT systems, including the SOQC combined with the new second-order subdifferential condition and the SOQC combined with tilt stability of local minimizers. Furthermore, under $\mathcal{C}^{2}$-cone reducibility, we prove that the Lipschitz-like property of the reference KKT system is equivalent to its strong regularity. Finally, we demonstrate that the Lipschitz-like property is equivalent to the nonsingularity of the generalized Jacobian associated with the KKT system under a certain verifiable assumption. These results provide a unified and rigorous framework for analyzing stability and sensitivity of solutions to composite optimization problems, as well as for the design and justification of numerical algorithms.