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Local convergence analysis of the augmented Lagrangian method (ALM) is established for a large class of composite optimization problems with nonunique Lagrange multipliers under a second-order sufficient condition. We present a new second-order variational property called the semistability of second subderivatives and demonstrate that it is widely satisfied for numerous classes of functions, which is important for applications in constrained and composite optimization problems. Using the latter condition and a certain second-order sufficient condition, we are able to establish Q-linear convergence of the primal-dual sequence for an inexact version of the ALM for composite programs. Funding: Research of the first author is partially supported by Singapore National Academy of Science via SASEAF Programme under the grant RIE2025 NRF International Partnership Funding Initiative. Research of the second author is partially supported by the National Science Foundation under the grant DMS 2108546. 

This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems. 

Understanding the role that subgradients play in various second-order variational anal- ysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of polyhedral functions, i.e., functions with poly- hedral convex epigraphs, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdif- ferential of polyhedral functions ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. This allows us to characterize continuous differentiability of the proximal mapping and twice continuous differentiability of the Moreau envelope of polyhedral functions. We close the paper with proving the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions.