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In cloud-edge computing paradigms, the integration of edge servers and task offloading mechanisms has posed new challenges to developing task scheduling strategies. This paper proposes an efficient convergent firefly algorithm (ECFA) for scheduling security-critical tasks onto edge servers and the cloud datacenter. The proposed ECFA uses a probability-based mapping operator to convert an individual firefly into a scheduling solution, in order to associate the firefly space with the solution space. Distinct from the standard FA, ECFA employs a low-complexity position update strategy to enhance computational efficiency in solution exploration. In addition, we provide a rigorous theoretical analysis to justify that ECFA owns the capability of converging to the global best individual in the firefly space. Furthermore, we introduce the concept of boundary traps for analyzing firefly movement trajectories, and investigate whether ECFA would fall into boundary traps during the evolutionary procedure under different parameter settings. We create various testing instances to evaluate the performance of ECFA in solving the cloud-edge scheduling problem, demonstrating its superiority over FA-based and other competing metaheuristics. Evaluation results also validate that the parameter range derived from the theoretical analysis can prevent our algorithm from falling into boundary traps.