Complete Characterizations of Well-Posedness in Parametric Composite Optimization
Boris S. Mordukhovich, Peipei Tang, Chengjing Wang
TL;DR
This work provides complete characterizations of well-posedness for KKT systems arising in perturbed composite optimization problems of the form $\min_x h(x)+g(F(x))$, leveraging parabolic regularity to reduce second-order subdifferentials to a novel second-order variational function. It extends classical second-order sufficient conditions to the composite setting and establishes equivalences among the Aubin property, strong regularity, tilt stability, and nondegeneracy under $\mathcal{C}^{2}$-cone reducibility. By linking SOSC/SSOSC with the second-order subdifferential and tilt stability, the authors present a unified framework for stability, sensitivity analysis, and the justification of numerical algorithms for a broad class of constrained optimization problems. The results unify and extend prior theories across NLPs, conic programs, and generalized equations, with practical implications for algorithm design and robust 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.