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Centralized Network Utility Maximization with Accelerated Gradient Method

Ying Tian, Zhiliang Wang, Xia Yin, Xingang Shi, Jiahai Yang, Han Zhang

TL;DR

This work addresses centralized network utility maximization (NUM) in SDN-controlled networks by formulating a smooth NUM objective that enables the use of Nesterov's accelerated gradient method (AGM). By introducing a smooth $(oldsymbol{ extalpha},oldsymbol{ extxi})$-fair utility and a Softplus-based smooth penalty, the authors derive a smooth objective $V_S$ with Lipschitz gradient, achieving a convergence rate of $O(d/t^2)$ in the number of iterations $t$ and a per-iteration cost that scales with the number of flows $d$, i.e., $O( ext{cost per iteration})=O(d)$. They further enhance practical performance with two restart strategies improving convergence stability. Evaluations on realistic topologies show AGM and its restart variants converge faster and provide close-to-optimal network utility, demonstrating strong scalability to large $d$ and favorable efficiency compared to prior centralized methods like Exp-NUM. The approach provides a practical, scalable, and fast centralized NUM solution suitable for SDN controllers managing large-scale cloud and inter-datacenter networks.

Abstract

Network utility maximization (NUM) is a well-studied problem for network traffic management and resource allocation. Because of the inherent decentralization and complexity of networks, most researches develop decentralized NUM algorithms. In recent years, the Software Defined Networking (SDN) architecture has been widely used, especially in cloud networks and inter-datacenter networks managed by large enterprises, promoting the design of centralized NUM algorithms. To cope with the large and increasing number of flows in such SDN networks, existing researches about centralized NUM focus on the scalability of the algorithm with respect to the number of flows, however the efficiency is ignored. In this paper, we focus on the SDN scenario, and derive a centralized, efficient and scalable algorithm for the NUM problem. By the designing of a smooth utility function and a smooth penalty function, we formulate the NUM problem with a smooth objective function, which enables the use of Nesterov's accelerated gradient method. We prove that the proposed method has $O(d/t^2)$ convergence rate, which is the fastest with respect to the number of iterations $t$, and our method is scalable with respect to the number of flows $d$ in the network. Experiments show that our method obtains accurate solutions with less iterations, and achieves close-to-optimal network utility.

Centralized Network Utility Maximization with Accelerated Gradient Method

TL;DR

This work addresses centralized network utility maximization (NUM) in SDN-controlled networks by formulating a smooth NUM objective that enables the use of Nesterov's accelerated gradient method (AGM). By introducing a smooth -fair utility and a Softplus-based smooth penalty, the authors derive a smooth objective with Lipschitz gradient, achieving a convergence rate of in the number of iterations and a per-iteration cost that scales with the number of flows , i.e., . They further enhance practical performance with two restart strategies improving convergence stability. Evaluations on realistic topologies show AGM and its restart variants converge faster and provide close-to-optimal network utility, demonstrating strong scalability to large and favorable efficiency compared to prior centralized methods like Exp-NUM. The approach provides a practical, scalable, and fast centralized NUM solution suitable for SDN controllers managing large-scale cloud and inter-datacenter networks.

Abstract

Network utility maximization (NUM) is a well-studied problem for network traffic management and resource allocation. Because of the inherent decentralization and complexity of networks, most researches develop decentralized NUM algorithms. In recent years, the Software Defined Networking (SDN) architecture has been widely used, especially in cloud networks and inter-datacenter networks managed by large enterprises, promoting the design of centralized NUM algorithms. To cope with the large and increasing number of flows in such SDN networks, existing researches about centralized NUM focus on the scalability of the algorithm with respect to the number of flows, however the efficiency is ignored. In this paper, we focus on the SDN scenario, and derive a centralized, efficient and scalable algorithm for the NUM problem. By the designing of a smooth utility function and a smooth penalty function, we formulate the NUM problem with a smooth objective function, which enables the use of Nesterov's accelerated gradient method. We prove that the proposed method has convergence rate, which is the fastest with respect to the number of iterations , and our method is scalable with respect to the number of flows in the network. Experiments show that our method obtains accurate solutions with less iterations, and achieves close-to-optimal network utility.
Paper Structure (21 sections, 12 theorems, 37 equations, 4 figures, 4 tables, 4 algorithms)

This paper contains 21 sections, 12 theorems, 37 equations, 4 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Network Utility Maximization Problem
  4. Problem Formulation
  5. Penalty-based Formulation
  6. Gradient Methods for NUM with The Exact Penalty Function
  7. NUM Problem with Smooth Objective Function
  8. Preliminaries
  9. Smooth Utility Function
  10. Smooth Penalty Function
  11. NUM with Smooth Objective Function
  12. Accelerated Gradient Method for NUM with Smooth Objection Function
  13. Vanilla Accelerated Gradient Method
  14. Accelerated Gradient Method with Restart
  15. Evaluation
  16. ...and 6 more sections

Key Result

Proposition 1

If Algorithm pgd is run for $T$ iterations with initial point $\bm{x}^{(0)}$, step size $\gamma = \frac{\Vert \bm{x}^{*}\!-\bm{x}^{(0)} \Vert }{ M\sqrt{dt}}$, then the solution $\hat{\bm{x}}^{(T)}$ satisfies: i.e., the error is $\varepsilon= O(\sqrt{d}/\sqrt{t})$ at step $t$, or equivalently, PGD achieves an $\varepsilon$-optimal solution with $\Omega(d/\varepsilon^2)$ iterations.

Figures (4)

  • Figure 1: Graphs of ReLU function and Softplus function.
  • Figure 2: The objective value and error at each iteration in GÉANT and RF1221.
  • Figure 3: The network utility value and error at each iteration when $U_s=\sum_{s\in S}x_s$ in GÉANT and RF1221.
  • Figure 4: The error at each iteration with different number of flows $d$ when $U_s=\sum_{s\in S}\ln{(x_s+0.5)}$ in GÉANT.

Theorems & Definitions (21)

  • Proposition 1: Convergence rate of PGDMAL-050
  • Proposition 2: Convergence rate of Exp-NUM8737600
  • Definition 1: Lipschitz continuous
  • Lemma 1
  • Lemma 2
  • Definition 2: Smooth
  • Lemma 3
  • Lemma 4
  • proof
  • Remark
  • ...and 11 more