Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 984047, 10 pages

http://dx.doi.org/10.1155/2015/984047

## A Cooperative Optimization Algorithm Inspired by Chaos–Order Transition

^{1}School of Computer Science and Technology, Huaibei Normal University, Huaibei 235000, China^{2}College of Information Engineering, Chaohu University, Chaohu 238000, China

Received 17 May 2015; Revised 29 July 2015; Accepted 3 August 2015

Academic Editor: Matteo Gaeta

Copyright © 2015 Fangzhen Ge et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The growing complexity of optimization problems in distributed systems (DSs) has motivated computer scientists to strive for efficient approaches. This paper presents a novel cooperative algorithm inspired by chaos–order transition in a chaotic ant swarm (CAS). This work analyzes the basic dynamic characteristics of a DS in light of a networked multiagent system at microlevel and models a mapping from state set to self-organization mechanism set under the guide of system theory at macrolevel. A collaborative optimization algorithm (COA) in DS based on the chaos–order transition of CAS is then devised. To verify the validity of the proposed model and algorithm, we solve a locality-based task allocation in a networked multiagent system that uses COA. Simulations show that our algorithm is feasible and effective compared with previous task allocation approaches, thereby illustrating that our design ideas are correct.

#### 1. Introduction

Distributed system (DS) is composed of many distributed computational components/units that mutually communicate and cooperate in networks. Owing to developments in computer networks and communication and distributed programming technology, DSs have become extensive. Ad hoc networks [1], peer-to-peer systems [2], and pervasive computing systems [3] are various types of DS. In general, DSs have a few common features as follows.

*(1) Decentralized Control*. DSs have a large scale and dynamical structure, so entire systems do not have a global control node. The actions of nodes are random. An example is the routing protocol for resource-constraint opportunistic networks [4], in which the intermittent connected nodes transfer messages from one node to another based on dynamic topology.

*(2) Sharing of Local Resources*. In this paper, resources refer to the buffer of nodes in opportunistic networks, replicas of popular web objects, computational capacity of nodes in wireless sensor networks, and so forth. All these resources are distributed among a group of nodes and are shared by these nodes [5].

*(3) Openness, Dynamics, and Unreliability of DS*. Most DSs are open with a dynamic structure, so the link of nodes may be unreliable [6]. Moreover, due to the autonomy of nodes, each node always communicates with its neighboring nodes to maximize the total utilities [7].

*(4) Constraints of DS*. In a DS, each task will be executed on available nodes. However, in a real network, the resources of each node are limited, such as energy, channel, and bandwidth. Sometimes a variety of conditions exist in a DS, numerous relationships are intertwined (including the relationship between nonlinear dynamics and unknown mechanisms), and “relationships” are difficult to decompose into a combination of a number of simple relationships.

Recently, DSs received considerable attention because of their large number of applications, such as task and resource allocation in grids, performance optimization in multi-agent-based e-commerce systems, data fusion in distributed sensor networks, and logistics in enterprise resource planning. However, the application requirements of DSs suffer from the aforementioned four common features. Traditional optimization methods often fail in terms of rational allocation of resources and optimal performance of DS. For instance, Asynchronous Distributed Constraint Optimization [8] and Optimal Asynchronous Partial Overlay [9] require a global topological structure to ensure optimal performance. However, acquiring global topology in DS is difficult or even impossible.

Cooperative optimization approaches of DSs have attracted the attention of many scholars. Existing methods can be divided into two categories. The first category is designing a distributed algorithm. The objective function of a DS is the sum of the node’s functions. The function value of each node is acquired by exchanging with its neighbors. Bertsekas and Tsitsiklis [10, 11] put forward a parallel and distributed computing theory frame for a set of processor scheduling problems. Nedić and Ozdaglar [12] propose an average subgradient consensus algorithm to solve the convex optimization problem of a DS with constraints and further present an extended subgradient algorithm in light of the constraints on a set of nodes. Srivastava and Nedić [13, 14] devise a distributed optimization algorithm with asynchronous communication. Zhu and Martinez [15] use the primal dual subgradient algorithm to solve the optimization problem with constraints, which are described by a set of global equality and inequality. In summary, these studies are closely related with the networked utility maximization optimization [16, 17]. The second category is heuristic algorithms based on a specific behavior. Consensus-based decentralized auction algorithm (CBAA) [18] is a distributed task allocation algorithm based on auction and consistency, where each iteration runs the auction and consensus, and the algorithm uniformly converges to the optimal solution. Market-based algorithm (MBA) [19] is a distributed task allocation algorithm based on the market mechanism, where each agent can buy a task, trade transactions among them, and minimize the purchase price. Swarm-GAP (SGAP) [20] is an approximate algorithm for distributed task allocation based on the principle of division of labor in social insects, where an individual selects a task based on the stimuli of the task and their preference to the task.

However, none of the aforementioned studies consider the autonomy of the individual, especially self-organization, self-adaptation, and other advanced functions in DSs. Consequently, these algorithms cannot embody the advanced features of DSs. The current paper aims to employ the advanced functions and construct a distributed cooperative optimization algorithm (COA).

Importantly, the thinking of chaos–order transition in the foraging behavior of ants [21] gives us a vital spark of imagination for solving cooperative optimization problems. The entire foraging process of chaotic ants is controlled by three successive strategies, namely, hunting, homing, and path building. During this process, the behavior of ants transforms from chaos to order under the influence of the ants’ intelligence and experience.

This paper investigates the dynamic characteristics of an agent with autonomy at the microlevel and devises a cooperative optimization model for DSs at the macrolevel. On the basis of our studies on chaotic ant swarm (CAS) [22–24], we propose a COA. This paper focuses on various perspectives as follows.

The autonomy of agents in DSs plays a key role in representing the functions of DSs, where autonomy means that agents have spontaneous, initiative behaviors.

A mechanism is shown to bridge the microlevel and macrolevel.

Specifically, our main contributions are summarized as follows.

Our collaborative optimization model can be applied to describe the dynamic evolution of self-organization in DS.

Our algorithm is parallel because the individuals in the proposed algorithm evolve dynamically and are parallel with little scale relations.

Our algorithm fully embodies the autonomy of individual and interactive systems.

The remainder of the paper is organized as follows. Section 2 analyzes the basic dynamic characteristics of an autonomous agent in a multiagent network from the point of view of dynamics and explores the collaborative optimization model of DS. Section 3 describes how a CAS can be used to derive an effective COA. Section 4 depicts the application to a locality-based task allocation. Finally, Section 5 concludes our paper and provides further research content.

#### 2. Cooperative Optimization Model

To construct our cooperative optimization model, we first analyze the dynamic characteristics at the micro scope and then devise the mapping method at the macro scope.

##### 2.1. Dynamic Characteristics of an Autonomous Agent

Many researchers use the networked multiagent system based on local information structures to study DSs and have achieved good results [25, 26]. We also apply the networked multiagent system to analyze the dynamic characteristics of DSs and to set up the general dynamic characteristics of an agent.

The networked multiagent system can be expressed by a graph, where the vertex of the graph denotes an agent, while the edge represents the relationship between two agents. A networked multiagent system is shown in Figure 1, where agent is under the influence of agent .