An Accelerated Distributed Optimization with Equality and Inequality Coupling Constraints

TL;DR

The paper tackles distributed convex optimization with both affine equality and nonlinear inequality couplings by leveraging a dual formulation that yields a consensus problem over a network. It introduces an accelerated linearized method of multipliers that uses look-ahead gradients and a Laplacian-based penalty to achieve faster, nonergodic convergence for both optimality and feasibility. Theoretical results establish O(1/N^2) + O(1/N) rates, and empirical tests on a 20-node network show the method outperforms ALT and dual subgradient baselines under the same communication budget. This approach enables efficient, scalable decentralized coordination in systems with global coupling constraints, such as smart grids and distributed resource allocation scenarios.

Abstract

This paper studies distributed convex optimization with both affine equality and nonlinear inequality couplings through the duality analysis. We first formulate the dual of the coupling-constraint problem and reformulate it as a consensus optimization problem over a connected network. To efficiently solve this dual problem and hence the primal problem, we design an accelerated linearized algorithm that, at each round, a look-ahead linearization of the separable objective is combined with a quadratic penalty on the Laplacian constraint, a proximal step, and an aggregation of iterations. On the theory side, we prove non-ergodic rates for both the primal optimality error and the feasibility error. On the other hand, numerical experiments show a faster decrease of optimality error and feasibility residual than augmented-Lagrangian tracking and distributed subgradient baselines under the same communication budget.

Figures (2)

• Figure 1: The primal optimal error under the four algorithms.
• Figure 2: The absolute error of the violation of constraints under the four algorithms.

Theorems & Definitions (4)

• Proposition 1
• proof
• Theorem 1
• proof