Distributed Random-Fixed Projected Algorithm for Constrained Optimization Over Digraphs

TL;DR

This paper introduces a novel distributed optimization algorithm for directed graphs that does not require the graph to be balanced, using an epigraph reformulation and recursive updates to ensure convergence.

Contribution

It proposes a new two-step recursive algorithm for constrained optimization over unbalanced digraphs, relaxing previous graph assumptions.

Findings

Algorithm converges asymptotically to a common optimal solution.

Effective in strongly connected digraphs, demonstrated by simulations.

Removes the need for balanced or doubly-stochastic graphs.

Abstract

This paper is concerned with a constrained optimization problem over a directed graph (digraph) of nodes, in which the cost function is a sum of local objectives, and each node only knows its local objective and constraints. To collaboratively solve the optimization, most of the existing works require the interaction graph to be balanced or "doubly-stochastic", which is quite restrictive and not necessary as shown in this paper. We focus on an epigraph form of the original optimization to resolve the "unbalanced" problem, and design a novel two-step recursive algorithm with a simple structure. Under strongly connected digraphs, we prove that each node asymptotically converges to some common optimal solution. Finally, simulations are performed to illustrate the effectiveness of the proposed algorithms.