In this paper, we study the inexact Moreau envelope Lagrangian (iMELa) method for solving smooth non-convex optimization problems over a simple polytope with additional convex inequality constraints. By incorporating a proximal term into the traditional Lagrangian function, the iMELa method approximately solves a convex optimization subproblem over the polyhedral set at each main iteration. Under the assumption of a local error bound condition for subsets of the feasible set defined by subsets of the constraints, we establish that the iMELa method can find an $\epsilon$-Karush-Kuhn-Tucker point with $\tilde O(\epsilon^{-2})$ gradient oracle complexity.

Federated learning (FL) enables multiple clients with distributed data sources to collaboratively train a shared model without compromising data privacy. However, existing FL paradigms face challenges due to heterogeneity in client data distributions and system capabilities. Personalized federated learning (pFL) has been proposed to mitigate these problems, but often requires a shared model architecture and a central entity for parameter aggregation, resulting in scalability and communication issues. More recently, model-heterogeneous FL has gained attention due to its ability to support diverse client models, but existing methods are limited by their dependence on a centralized framework, synchronized training, and publicly available datasets. To address these limitations, we introduce Federated Peer-Adaptive Ensemble Learning (FedPAE), a fully decentralized pFL algorithm that supports model heterogeneity and asynchronous learning. Our approach utilizes a peer-to-peer model sharing mechanism and ensemble selection to achieve a more refined balance between local and global information. Experimental results show that FedPAE outperforms existing state-of-the-art pFL algorithms, effectively managing diverse client capabilities and demonstrating robustness against statistical heterogeneity.

We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations.