Recommender systems play a pivotal role across practical scenarios, showcasing remarkable capabilities in user preference modeling. However, the centralized learning paradigm predominantly used raises serious privacy concerns. The federated recommender system (FedRS) addresses this by updating models on clients, while a central server orchestrates training without accessing private data. Existing FedRS approaches, however, face unresolved challenges, including non-convex optimization, vulnerability, potential privacy leakage risk, and communication inefficiency. This paper addresses these challenges by reformulating the federated recommendation problem as a convex optimization issue, ensuring convergence to the global optimum. Based on this, we devise a novel method, RFRec, to tackle this optimization problem efficiently. In addition, we propose RFRecF, a highly efficient version that incorporates non-uniform stochastic gradient descent to improve communication efficiency. In user preference modeling, both methods learn local and global models, collaboratively learning users' common and personalized interests under the federated learning setting. Moreover, both methods significantly enhance communication efficiency, robustness, and privacy protection, with theoretical support. Comprehensive evaluations on four benchmark datasets demonstrate RFRec and RFRecF's superior performance compared to diverse baselines.