OPTIMIZATION ALGORITHMS FOR LARGE-SCALE DATA SYSTEMS: A MATHEMATICAL FRAMEWORK FOR COMPUTATIONAL EFFICIENCY

Abstract

Large-scale data systems strain traditional optimization techniques because modern workloads involve high-dimensional variables, heterogeneous storage layers, distributed compute nodes, and rapidly shifting resource states. This paper develops a consolidated mathematical framework that unifies algorithmic optimization methods with system-level constraints to improve computational efficiency in such environments. The framework integrates convex optimization, stochastic gradient methods, distributed primal–dual updates, and adaptive scheduling models, allowing each component of a data system memory, I/O, communication bandwidth, task queues, and compute throughput to be expressed as an optimization surface. By formalizing computation cost as a multivariate function of latency, bandwidth contention, node failures, and parallelization overhead, the framework enables analytical comparisons of algorithmic strategies under real system conditions. The study synthesizes existing optimization models and proposes a hybrid large-scale solver that combines coordinate descent, variance-reduced gradients, and asynchronous update rules to stabilize convergence even when data is partitioned unevenly or updates are delayed. The resulting formulation reduces communication rounds, improves throughput, and demonstrates strong resilience to straggler effects. This work contributes a mathematically grounded perspective bridging optimization theory with distributed systems engineering, offering a foundation for scalable, efficient computation in data-intensive infrastructures.