Operator equation formulation and solution analysis for distributed resource allocation in large-scale infrastructure networks

Abstract

Resource allocation in modern large-scale infrastructure networks, such as cloud edge-computing grids, smart power distribution systems, and dynamic multi-agent wireless systems, relies on decentralized, scalable, and communication-efficient decision-making; however, centralized optimization does not solve the issue with networks of different sizes and creates single-point failures. The article introduces the operator-theoretic framework - a unified method for reformulating the problem of constrained distributed resource allocation in complex network topologies as the problem of finding a zero of a monotonic operator in a real Hilbert space. The physical topology constraints are expressed using graph Laplacians, while resource limitations are expressed in the form of indicator functions, which allows transforming the complex global allocation problem into the operator equation 0 = (A+B)(x), where A represents the subdiffential of the local cost and constraints structure, while B outlines the interconnections of the network involved. The theory of maximal monotonic operators, Minty’s theorem, and the resolvent’s firm non-expansiveness help establish the existence, uniqueness, and metrics of the solution. A distributed Douglas-Rachford splitting algorithm is developed afterwards; its global weak and strong convergence is verified through the Krasnoselskii-Mann theorem for averaged operators. At the same time, explicit linear convergence rate under strong monotonicity conditions is provided. Numerical simulations, implemented for synthetic scale-free and small-world network topologies with the number of nodes provided (N=10,000), demonstrate that the proposed approach guarantees the getting accuracy of 1e-6 within 3.85 seconds, at the same time when the centralized interior point algorithm provides no solutions due to memory shortage and completes the problem successfully in 0.03-1.10 seconds with N<10,000. The communication overhead per node does not exceed approximately 6.2 messages per iteration regardless of the network size, while it increases in proportion to N for the centralized approach. These results demonstrate that the proposed operator-equation framework provides mathematically rigorous convergence guarantees together with practically verified scalability for decentralized resource allocation in large-scale infrastructure networks.