Title: Projected Primal-Dual Gradient Flow of Augmented Lagrangian with Application to Distributed Maximization of the Algebraic Connectivity of a Network,
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.