Mathematical modeling and optimization in complex systems: a review of methods and emerging trends

Abstract

Complex systems — characterized by nonlinear relationships, multiple interacting scales, and emergent behavior — represent one of the central challenges facing contemporary engineering, applied physics, and computational science; understanding and controlling their dynamics requires mathematical tools capable of representing uncertainty, structural interdependence, and the temporal evolution of their components. This narrative review aims to synthesize mathematical modeling methods and optimization techniques applied to complex systems, as well as to identify the emerging trends that are redefining the field. The literature search was carried out in indexed scientific databases, combining Spanish and English terms related to mathematical modeling, optimization, metaheuristics, digital twins, and complex network theory, prioritizing publications from the last five years; this process allowed for the examination of a broad set of studies and the selection of twenty-two relevant references. The results show a consolidation of bio-inspired metaheuristic algorithms as the dominant tool for non-convex optimization problems, sustained progress in hybrid models that combine physical simulation with machine learning — particularly in digital twins — and a growing application of complex network theory to represent the structural interdependence of engineering systems. It is concluded that the convergence of artificial intelligence, multi-objective optimization, and network-based modeling constitutes the most promising trajectory for addressing the growing complexity of contemporary systems, although challenges remain regarding interpretability, computational cost, and the empirical validation of proposed models.