Primal-Dual Algorithm for Distributed Constrained Optimization

TL;DR

This paper introduces a distributed primal-dual algorithm based on the augmented Lagrange method for solving constrained optimization problems across multiple agents, ensuring consensus and convergence to optimal solutions.

Contribution

It proposes a novel distributed primal-dual algorithm with projection, achieving consensus and convergence with rate analysis, distinct from existing methods.

Findings

Agents reach consensus asymptotically.

Cost function value converges at rate O(1/k).

Algorithm outperforms existing distributed constrained optimization methods.

Abstract

The paper studies a distributed constrained optimization problem, where multiple agents connected in a network collectively minimize the sum of individual objective functions subject to a global constraint being an intersection of the local constraint sets assigned to the agents. Based on the augmented Lagrange method, a distributed primal-dual algorithm with a projection operation included is proposed to solve the problem. It is shown that with appropriately chosen constant step size, the local estimates derived at all agents asymptotically reach a consensus at an optimal solution. In addition, the value of the cost function at the time-averaged estimate converges with rate $O(\frac{1}{k})$ to the optimal value for the unconstrained problem. By these properties the proposed primal-dual algorithm is distinguished from the existing algorithms for distributed constrained optimization. The theoretical analysis is justified by numerical simulations.