The mathematical modeling and numerical simulation of sustainable energy systems present fundamental challenges in optimization theory, requiring rigorous treatment of stability, convergence, and computational efficiency in large-scale applied mathematical frameworks. This review examines the mathematical foundations of optimization algorithms and numerical simulation techniques employed in sustainable energy system modeling, with emphasis on convergence analysis, stability characteristics, and computational complexity considerations. The increasing penetration of renewable energy sources necessitates sophisticated mathematical formulations that capture both deterministic and stochastic behaviors in power networks, storage systems, and smart grid infrastructures. Core optimization methodologies including convex programming, nonlinear optimization, stochastic programming, and numerically-guaranteed metaheuristic approaches are analyzed through the lens of convergence theory and stability analysis. Particular attention is devoted to decomposition methods, interior-point algorithms, and Newton-type methods that demonstrate provable convergence properties for large-scale energy system applications. The mathematical formulation of power flow equations, storage dynamics, and network constraints requires careful consideration of numerical stability in time-dependent simulations. Computational efficiency considerations encompassing scalability, parallelization capabilities, and error control mechanisms are examined within the context of sparse linear algebra and high-performance computing frameworks. The review concludes by identifying emerging directions in robust optimization under uncertainty, adaptive numerical methods, and reduced-order modeling techniques that promise to advance the mathematical foundations of sustainable energy system optimization.