Nguyen Thi Hoai Linh

FLOCKING AND NON-FLOCKING BEHAVIOR IN A STOCHASTIC CUCKER–SMALE SYSTEM

We first present a new stochastic version of the Cucker-Smale model of the emergent behavior in flocks in which the mutual communication between individuals is affected by random factor. Then, the existence and uniqueness of global solution to this system are verified. We show a result which agrees with natural fact that under the effect of large noise, there is no flocking. In contrast, if noise is small, then flocking may occur. Paper ends with some numerical examples.

DYNAMICS OF A STOCHASTIC RATIO-DEPENDENT PREDATOR-PREY MODEL

In this paper, we consider a stochastic ratio-dependent predator-prey model. We firstly prove the existence, uniqueness and positivity of the solutions. Then, the boundedness of moments of population are studied. Finally, we show the upper-growth rates and exponential death rates of population under some conditions.

Distributed constrained optimization protocol via an exact penalty method

The purpose of this paper is to propose a protocol for distributed multi-agent optimization problem to minimize the average of objective functions of the agents in the network with satisfying constraints of each agent. The exact penalty method is applied to distributed optimization via a linear protocol, with employing two step-size parameters for the objective function and the constraint function of each agent. The proposed protocol works only with the decision variables and does not need additional variables such as dual variables. A proof of the convergence of the proposed protocol is provided as well as the boundedness under mild assumptions. The protocol is also illustrated by a numerical example.

Distributed optimization with equality and inequality constraints with delayed information of feasibility

This paper proposes a protocol for a distributed optimization problem in multi-agent networks with equality and inequality constraints. Instead of computing dual optimizations as in previous protocols that can handle constraints, the proposed protocol utilizes additional data of information on past fulfillment of the constraints. Since equality constraints no longer enjoy techniques exploiting strict feasibility of inequality constraints, problems with equality constraints can never be solved by simply extending methods for inequality constraints. To develop a protocol, this paper introduces two diminishing parameters, one of which controls step sizes of decision variables moving to an optimum, while the other specifies the error bound of the equality constraints in each step of iteration. Appropriate choices of these parameters lead to a proof of consensus and convergence of the proposed protocol. A computational example is provided that illustrates the new protocol.

Convergence time evaluation of a gossip algorithm over signed graphs

Convergence time is investigated for a gossip algorithm over a connected signed graph, where each edge of the graph has positive or negative sign. The algorithm is an iterative procedure. At each time, (i) two nodes directly connected with an edge are chosen randomly, (ii) they exchange their values according to the sign of the edge, and (iii) they update their values as the average of each nodes value and its received value. It is shown that the values of the algorithm always converge in mean square, where a bipartite consensus or a trivial consensus is achieved. A convergence time is defined as the smallest time such that it takes for the values of the algorithm to get within a given neighborhood of the consensus value with high probability, regardless of initial state. An upper bound of the convergence time is given in terms of a characteristic value of the given graph.

Opinion formation under bounded confidence via gossip algorithms

This paper investigates two bounded confidence gossip algorithms, one with constant confidence threshold and the other with increasing one, for effective communicating between agents in a network among whom some opinion formation forms. Each agent in the network keeps a real value presenting its opinion about some matter. The opinions of agents will be updated time by time according to an iterative procedure. At each time, (i) two arbitrary agents are chosen randomly, (ii) they exchange their opinions, and (iii) if the distance between the opinions does not exceed some given confidence threshold, they update their opinions as the average of the two. It is shown that the algorithms almost surely drive any initial opinion profile to some opinion profile in which any two opinions either are the same or differ more than the confidence threshold. Moreover, the second algorithm can help achieving a prescribed number of different opinions on the convergence opinion profiles.