# Projected Primal-Dual Gradient Flow of Augmented Lagrangian with Application to Distributed Maximization of the Algebraic Connectivity of a Network

## TL;DR

This paper introduces a projected primal-dual gradient flow method for convex optimization with constraints, demonstrating its convergence and application to distributed network connectivity maximization through SDP relaxation.

## Contribution

It develops a novel projected primal-dual gradient flow approach for convex problems with constraints and applies it to distributed algebraic connectivity maximization.

## Key findings

- The proposed method converges to saddle points and optimal solutions.
- Numerical examples confirm convergence and effectiveness.
- The relaxation from SDP to NP is empirically validated.

## Abstract

In this paper, a projected primal-dual gradient flow of augmented Lagrangian is presented to solve convex optimization problems that are not necessarily strictly convex. The optimization variables are restricted by a convex set with computable projection operation on its tangent cone as well as equality constraints. As a supplement of the analysis in \cite{niederlander2016distributed}, we show that the projected dynamical system converges to one of the saddle points and hence finding an optimal solution. Moreover, the problem of distributedly maximizing the algebraic connectivity of an undirected network by optimizing the port gains of each nodes (base stations) is considered. The original semi-definite programming (SDP) problem is relaxed into a nonlinear programming (NP) problem that will be solved by the aforementioned projected dynamical system. Numerical examples show the convergence of the aforementioned algorithm to one of the optimal solutions. The effect of the relaxation is illustrated empirically with numerical examples. A methodology is presented so that the number of iterations needed to reach the equilibrium is suppressed. Complexity per iteration of the algorithm is illustrated with numerical examples.