  This paper proposes an adaptive primal-dual dynamics for distributed
optimization in multi-agent systems. The proposed dynamics incorporates an
adaptive synchronization law that reinforces the interconnection strength
between the primal variables of the coupled agents, the given law accelerates
the convergence of the proposed dynamics to the saddle-point solution. The
resulting dynamics is represented as a feedback interconnected networked system
that proves to be passive. The passivity properties of the proposed dynamics
are exploited along with the LaSalle's invariance principle for hybrid systems,
to establish asymptotic convergence and stability of the saddle-point solution.
Further, the primal dynamics is analyzed for the rate of convergence and
stronger convergence bounds are established, it is proved that the primal
dynamics achieve accelerated convergence under the adaptive synchronization.
The robustness of the proposed dynamics is quantified using L2-gain analysis
and the correlation between the rate of convergence and robustness of the
proposed dynamics is presented. The effectiveness of the proposed dynamics is
demonstrated by applying it to solve distributed least squares and distributed
support vector machines problems.
