Title A Unified Framework for Robust and Numerically Stable BFGS Methods with Applications in Mobile Localization Type Presentation Keywords BFGS, Robust Optimization, Global Convergence, Mobile Localization, Time of Arrival, Nonlinear Optimization. Abstract This paper presents a comprehensive unified framework that addresses the fundamental limitations of classical BFGS1 methods in unconstrained nonlinear optimization. We synergistically integrate Yang’s robust BFGS approach with Gill and Runnoe’s factored self- scaled BFGS methodology, resulting in the RFSS-BFGS2 algorithm.The proposed framework is rigorously evaluated through practical mobile localization based on ToA3 measurements Extensive numerical experiments demonstrate that RFSS-BFGS achieves superior performance with enhanced convergence reliability and accelerated convergence rates compared to state-of-the-art variants. The algorithm provides accurate mobile localization with significantly improved positioning accuracy and computational efficiency compared to conventional approaches. Researchers poria keshtkar (First researcher) , Heidar Keshavarz (Second researcher)