## 1 Introduction  
Optimization is a rapidly developing field due to its practical application to many real life problems. In this paper, we study a special class of optimization problem in the form of:  
\[\min f(\mathbf{x})\text{, s.t. }\mathbf{L}\leq\mathbf{x}\leq\mathbf{U} \tag{1}\]  
where \(f(\mathbf{x})\) is the objective function to be minimized, \(\mathbf{x}=(x_{1},\,x_{2},\,\ldots,\,x_{n})\in R^{n}\) is the variable vector, and \(\mathbf{L}=(l_{1},\,l_{2},\,\ldots,\,l_{n})\) and \(\mathbf{U}=(u_{1},\,u_{2},\,\ldots,\,u_{n})\) are the lower bound and upper bound of \(\mathbf{x}\), respectively. That is, \(l_{i}\leq x_{i}\leq u_{i}\) for \(i=1\) to \(n\).  
When the objective function \(f(\mathbf{x})\) is nonlinear and has many variables (_i.e._, \(n\) is large), problem (1) becomes hard to solve directly by conventional mathematical methods, such as gradient descent. To resolve this difficulty, evolutionary algorithms based on meta-heuristics have been applied to solve problem (1). Some notable meta-heuristics include Genetic Algorithm [1,2], Particle Swarm Optimization [3], Electromagnetism-like Algorithm [4], Artificial Immune System [5], Artificial Bee Colony Algorithm [6,7], Ant Colony Optimization [8] and Differential Evolution Algorithm [9]. Imperialist Competitive Algorithm (ICA) is a new evolutionary algorithm based on the meta-heuristic of human's socio-political evolution. It was proposed by [PERSON] and [PERSON] in 2007 [10], and since its inception, it has been applied to solve many optimization problems in scheduling [11,12,13,14,15], classification [16,17] and machinery design [18,19,20,21,22,23,24,25].  
ICA is a population-based method, whose population contains a set of solutions, called _countries_. ICA divides its population of countries into several sub-populations, called _empires_. Within each empire, ICA moves all non-best countries (called _colonies_) in an empire toward the best country (called _imperialist_) in the same empire through an operation called _assimulation_, which can be regarded as a primitive form of Particle Swarm Optimization [26]. The feature that distinguishes ICA from other evolutionary algorithms is that ICA allows all empires to interact via competition with each other. The competition operation of ICA simply moves a colony from the weakest empire to another empire. From this aspect, ICA resembles Island Model Genetic Algorithm [27]. Overall, ICA is essentially an integration of Island Model and a primitive form of PSO.  
The impact of the competition operation is weaker than that of the assimilation operation since in each generation only one colony is directly affected by the competition operation but all colonies are moved by the assimilation operation. Previous work on ICA focuses mostly on improving the assimilation operation or replacing the assimilation operation with more powerful meta-heuristics, but none focuses on improving the competition operation or, more generally, on enhancing the interaction among the empires in ICA. The quality of interaction via the competition operation is poor since the interaction only involves moving an inferior solution (_i.e._, a colony from the weakest empire) to another empire. The objective of this paper is to enhance the interaction among the empires in ICA. Two methods are proposed to achieve this objective. The first method utilizes all imperialists to create a new artificial imperialist, and the second method uses a crossover operation on the imperialists to allow effective interaction among all empires. Performance study using a set of well-known benchmark functions supports the effectiveness of the proposed methods.  
The rest of this paper is organized as follows. Section 2 reviews ICA and its variants. Section 3 describes our proposed methods. Section 4 gives experimental results. Section 5 concludes this paper and gives directions for future research.