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Inspired by the subgradient push method developed recently by Nedić et al. we present a distributed dual averaging push algorithm for constrained nonsmooth convex optimization over time-varying directed graph. Our algorithm combines the dual averaging method with the push-sum technique and achieves an $O(1/ \sqrt{k})$ convergence rate. Compared with the subgradient push algorithm, our algorithm, first, addresses the constrained problems, and, second, has a faster convergence rate, and, third, simplifies the convergence analysis. We also generalize the proposed algorithm so that input variables of subgradient oracles have guaranteed convergence. [ABSTRACT FROM AUTHOR]

This paper develops stochastic consentability of linear multiagent systems with time delays and multiplicative noises. First, the stochastic stability for stochastic differential delay equations driven by multiplicative noises is examined, and the existence of the positive definite solution for a class of generalized algebraic Riccati equations (GAREs) is established. Then, sufficient conditions are deduced for the mean square and almost sure consentability and stabilization based on the developed stochastic stability and GAREs. Consensus protocols are designed for linear multiagent systems with undirected and leader-following topologies. It is revealed that multiagent consentability depends on certain characterizing system parameters, including linear system dynamics, communication graph, channel uncertainties, and time delay of the deterministic term. It is shown that a second-order integrator multiagent system is unconditionally mean square and almost surely consentable for any given noise intensities and time delay, and that the mean square and almost sure consensus can be achieved by carefully choosing the control gain according to certain explicit conditions. [ABSTRACT FROM PUBLISHER]

Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $\delta$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of \delta$ for consensus is 1/2$ . Optimization on convergence rate of this protocol is further investigated. It is proved that the fastest convergence rate, which is the theoretic optimal rate among all controls, is of the order 1/t for the worst topologies, which are balanced and satisfy the extensible joint-connectivity condition. For practical implementation, certain open-loop control strategies are introduced to achieve consensus with a convergence rate of the same order as the fastest convergence rate. Furthermore, a consensus condition is derived for nonstationary and strongly correlated random topologies. The algorithms and consensus conditions are applied to distributed consensus computation of mobile ad-hoc networks; and their related critical exponents are derived from relative velocities of mobile agents for guaranteeing consensus. [ABSTRACT FROM PUBLISHER]