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A Unified Dual Consensus Approach to Distributed Optimization with Globally-Coupled Constraints

Zixuan Liu, Xuyang Wu, Dandan Wang, Jie Lu

TL;DR

This work tackles distributed convex optimization with globally-coupled constraints and nonsmooth objectives by introducing a unified dual-consensus framework, DUCA, and its proximal extension, Pro-DUCA. By approximating the method of multipliers on the dual problem and employing carefully chosen weight matrices, the algorithms enable fully decentralized updates without requiring a closed-form dual. They achieve an $O(1/k)$ convergence rate for both objective and feasibility and encompass many existing consensus-optimization methods as special cases, while also offering principled primal recovery guarantees. Numerical experiments demonstrate that DUCA and Pro-DUCA outperform several state-of-the-art approaches in objective accuracy and constraint satisfaction, highlighting practical benefits for distributed decision-making problems.

Abstract

This article explores distributed convex optimization with globally-coupled constraints, where the objective function is a general nonsmooth convex function, the constraints include nonlinear inequalities and affine equalities, and the feasible region is possibly unbounded. To address such problems, a unified DUal Consensus Algorithm (DUCA) and its proximal variant (Pro-DUCA) are proposed, which are unified frameworks that approximate the method of multipliers applied to the corresponding dual problem in no need of a closed-form dual objective. With varied parameter settings, DUCA and Pro-DUCA not only extend a collection of existing consensus optimization methods to solve the dual problem that they used to be inapplicable to, but also aid in offering new efficient algorithms to the literature. The proposed unified algorithms are shown to achieve $O(1/k)$ convergence rates in terms of optimality and feasibility, providing new or enhanced convergence results for a number of existing methods. Simulations demonstrate that these algorithms outperform several state-of-the-art alternatives in terms of objective and feasibility errors.

A Unified Dual Consensus Approach to Distributed Optimization with Globally-Coupled Constraints

TL;DR

This work tackles distributed convex optimization with globally-coupled constraints and nonsmooth objectives by introducing a unified dual-consensus framework, DUCA, and its proximal extension, Pro-DUCA. By approximating the method of multipliers on the dual problem and employing carefully chosen weight matrices, the algorithms enable fully decentralized updates without requiring a closed-form dual. They achieve an convergence rate for both objective and feasibility and encompass many existing consensus-optimization methods as special cases, while also offering principled primal recovery guarantees. Numerical experiments demonstrate that DUCA and Pro-DUCA outperform several state-of-the-art approaches in objective accuracy and constraint satisfaction, highlighting practical benefits for distributed decision-making problems.

Abstract

This article explores distributed convex optimization with globally-coupled constraints, where the objective function is a general nonsmooth convex function, the constraints include nonlinear inequalities and affine equalities, and the feasible region is possibly unbounded. To address such problems, a unified DUal Consensus Algorithm (DUCA) and its proximal variant (Pro-DUCA) are proposed, which are unified frameworks that approximate the method of multipliers applied to the corresponding dual problem in no need of a closed-form dual objective. With varied parameter settings, DUCA and Pro-DUCA not only extend a collection of existing consensus optimization methods to solve the dual problem that they used to be inapplicable to, but also aid in offering new efficient algorithms to the literature. The proposed unified algorithms are shown to achieve convergence rates in terms of optimality and feasibility, providing new or enhanced convergence results for a number of existing methods. Simulations demonstrate that these algorithms outperform several state-of-the-art alternatives in terms of objective and feasibility errors.

Paper Structure

This paper contains 26 sections, 14 theorems, 54 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem and Motivation
  3. Algorithm Development
  4. Review of Approximated Method of Multipliers (AMM)
  5. DUCA with Indirect Computation of \ref{['eq:AMM_y_update']}
  6. Pro-DUCA
  7. Distributed Implementations
  8. Single-Exchange Implementation
  9. Double-Exchange Implementation
  10. Connections to the Existing Works
  11. Methods for Consensus Optimization
  12. Methods for Problem \ref{['prob:CC']}
  13. Augmented Lagrangian Tracking FaP23 and Tracking-ADMM FNN20
  14. IPLUX WWL23
  15. DPMM GoZ23
  16. ...and 11 more sections

Key Result

Lemma 1

Suppose Assumption asp:problem_structure-asp:diagonal hold. Given ${\bf y}^k$ and ${\bf z}^k$, then there exist variables ${\bf y}^{k+1}\in\mathbb R^{N(m+p)}$, ${\bf x}^{k+1}\in X$ and ${\boldsymbol \sigma}^{k+1}\in\mathbb R^{N(m+p)}$ satisfying: ${\bf x}^{k+1}\in {\cal X}({\bf y}^{k+1})$, $-{\bolds

Figures (1)

  • Figure 1: Convergence performance of DUCA with different parameter settings and four alternative methods (left), and performance of Pro-DUCA with different values of $\alpha$ (right).

Theorems & Definitions (31)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2: WuL23
  • ...and 21 more