Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Globally-Constrained Decentralized Optimization with Variable Coupling

Dandan Wang, Xuyang Wu, Zichong Ou, Jie Lu

TL;DR

This work tackles decentralized convex optimization with both variable coupling and globally-coupled constraints, where local objectives and constraints depend on neighboring decisions. It introduces a decentralized projected primal-dual algorithm built on a gradient-projection step, a virtual-queue based dual update, and a primal-dual-primal mechanism, enabling fully distributed implementation. The authors prove $O(1/k)$ convergence in both feasibility and optimality under mild smoothness and Slater-type conditions and validate the approach with two numerical experiments, including a challenging coupled-variable scenario. The proposed method offers improved computational efficiency (projection-based primal updates and constant step-sizes) and robust performance in networks with varying connectivity, making it well-suited for applications in formation control and cooperative edge computing.

Abstract

Many realistic decision-making problems in networked scenarios, such as formation control and collaborative task offloading, often involve complicatedly entangled local decisions, which, however, have not been sufficiently investigated yet. Motivated by this, we study a class of decentralized optimization problems with a variable coupling structure that is new to the literature. Specifically, we consider a network of nodes collaborating to minimize a global objective subject to a collection of global inequality and equality constraints, which are formed by the local objective and constraint functions of the nodes. On top of that, we allow such local functions of each node to depend on not only its own decision variable but the decisions of its neighbors as well. To address this problem, we propose a decentralized projected primal-dual algorithm. It first incorporates a virtual-queue technique with a primal-dual-primal scheme, and then linearizes the non-separable objective and constraint functions to enable decentralized implementation. Under mild conditions, we derive $O(1/k)$ convergence rates for both objective error and constraint violations. Finally, two numerical experiments corroborate our theoretical results and illustrate the competitive performance of the proposed algorithm.

Globally-Constrained Decentralized Optimization with Variable Coupling

TL;DR

This work tackles decentralized convex optimization with both variable coupling and globally-coupled constraints, where local objectives and constraints depend on neighboring decisions. It introduces a decentralized projected primal-dual algorithm built on a gradient-projection step, a virtual-queue based dual update, and a primal-dual-primal mechanism, enabling fully distributed implementation. The authors prove convergence in both feasibility and optimality under mild smoothness and Slater-type conditions and validate the approach with two numerical experiments, including a challenging coupled-variable scenario. The proposed method offers improved computational efficiency (projection-based primal updates and constant step-sizes) and robust performance in networks with varying connectivity, making it well-suited for applications in formation control and cooperative edge computing.

Abstract

Many realistic decision-making problems in networked scenarios, such as formation control and collaborative task offloading, often involve complicatedly entangled local decisions, which, however, have not been sufficiently investigated yet. Motivated by this, we study a class of decentralized optimization problems with a variable coupling structure that is new to the literature. Specifically, we consider a network of nodes collaborating to minimize a global objective subject to a collection of global inequality and equality constraints, which are formed by the local objective and constraint functions of the nodes. On top of that, we allow such local functions of each node to depend on not only its own decision variable but the decisions of its neighbors as well. To address this problem, we propose a decentralized projected primal-dual algorithm. It first incorporates a virtual-queue technique with a primal-dual-primal scheme, and then linearizes the non-separable objective and constraint functions to enable decentralized implementation. Under mild conditions, we derive convergence rates for both objective error and constraint violations. Finally, two numerical experiments corroborate our theoretical results and illustrate the competitive performance of the proposed algorithm.
Paper Structure (21 sections, 9 theorems, 51 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 9 theorems, 51 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Examples of Applications
  4. A Multi-Vehicle Formation Stabilization Problem
  5. Task Offloading over Cooperative Mobile Edge Computing Networks
  6. Algorithm Development
  7. Problem Transformation
  8. Algorithm Design
  9. Decentralized Implementation
  10. Convergence Analysis
  11. Convergence Results
  12. Comparative Discussion
  13. Parameter Selections
  14. Numerical Experiments
  15. First Numerical Example
  16. ...and 6 more sections

Key Result

Proposition 1

Suppose Assumption asm:primalprob holds. Then, problem eq:probleminy satisfies the following properties:

Figures (3)

  • Figure 1: Network topology with $n=50$ nodes
  • Figure 2: Convergence performance of Algorithm \ref{['alg:algorithm']} and alternative methods for solving problem \ref{['eq:numercialexamp1']}.
  • Figure 3: Convergence performance of Algorithm \ref{['alg:algorithm']} for solving problem \ref{['eq:numercialexamp2']}.

Theorems & Definitions (22)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 12 more