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

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