Resource Allocation and Convergence Efficiency Optimization of Distributed Auto-Scaling Systems Mathematically

Core Focus: This paper presents a comprehensive mathematical framework to optimize resource utilization, efficiency, and operating costs in distributed cloud auto-scaling systems. Problem & Constraints: It defines a multi-dimensional objective function to simultaneously optimize CPU, memory, network bandwidth, and cost efficiency, enforced by hard operational constraints covering workload capacity, QoS response times, resource utilization bounds, and daily budgets. Key Findings: Evaluated across four operational scenarios (normal business hours, peak load, off-peak hours, and disaster recovery), the results show that Quasi-Newton BFGS offers the best practical balance—delivering high efficiency (~0.743 average) nearly three times faster than Newton's method while avoiding the slow, inefficient convergence of Steepest Descent.