Distributed Regularized Dual Gradient Algorithm for Constrained Convex Optimization over Time-Varying Directed Graphs

TL;DR

This paper introduces a distributed regularized dual gradient algorithm for constrained convex optimization over time-varying directed graphs, achieving fast convergence and explicit constraint violation bounds.

Contribution

It proposes a novel distributed algorithm based on push-sum and dual decomposition that handles time-varying directed networks without requiring compact dual sets.

Findings

Achieves convergence rate of O(ln T / T) for strongly convex objectives.

Provides explicit bounds on constraint violations.

Demonstrates efficiency through numerical experiments on network utility maximization.

Abstract

We investigate a distributed optimization problem over a cooperative multi-agent time-varying network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling constraints. Based on push-sum protocol and dual decomposition, we design a distributed regularized dual gradient algorithm to solve this problem, in which the algorithm is implemented in time-varying directed graphs only requiring the column stochasticity of communication matrices. By augmenting the corresponding Lagrangian function with a quadratic regularization term, we first obtain the bound of the Lagrangian multipliers which does not require constructing a compact set containing the dual optimal set when compared with most of primal-dual based methods. Then, we obtain that the convergence rate of the proposed method can achieve the order of $\mathcal{O}(\ln T/T)$ for strongly convex objective functions, where $T$ is the iterations. Moreover, the explicit bound of constraint violations is also given. Finally, numerical results on the network utility maximum problem are used to demonstrate the efficiency of the proposed algorithm.