Elsevier

Applied Soft Computing

Volume 29, April 2015, Pages 395-410
Applied Soft Computing

An immune multi-objective optimization algorithm with differential evolution inspired recombination

https://doi.org/10.1016/j.asoc.2015.01.012Get rights and content

Highlights

  • A DE inspired recombination operator is developed for continuous MOPs.

  • The proposed operator performs two complementary searching behaviors.

  • The proposed operator can be integrated with any MOEAs.

Abstract

According to the regularity of continuous multi-objective optimization problems (MOPs), an immune multi-objective optimization algorithm with differential evolution inspired recombination (IMADE) is proposed in this paper. In the proposed IMADE, the novel recombination provides two types of candidate searching directions by taking three recombination parents which distribute along the current Pareto set (PS) within a local area. One of the searching direction provides guidance for finding new points along the current PS, and the other redirects the search away from the current PS and moves towards the target PS. Under the background of the SBX (Simulated binary crossover) recombination which performs local search combined with random search near the recombination parents, the new recombination operator utilizes the regularity of continuous MOPs and the distributions of current population, which helps IMADE maintain a more uniformly distributed PF and converge much faster. Experimental results have demonstrated that IMADE outperforms or performs similarly to NSGAII, NNIA, PESAII and OWMOSaDE in terms of solution quality on most of the ten testing MOPs. IMADE converges faster than NSGAII and OWMOSaDE. The efficiency of the proposed DE recombination and the contributions of DE and SBX recombination to IMADE have also been experimentally investigated in this work.

Graphical abstract

This figure illustrates the two searching strategies of the proposed differential evolution inspired recombination operator: move towards the target Pareto set to improve the approximation and search along the current Pareto set to enhance the diversity.

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Introduction

Many optimization problems in real-life applications have more than one objective that is in conflict with the others. Such problems are termed as multi-objective problems (MOPs). This paper considers the following continuous MOPs:

minF(x)=(f1(x),f2(x),,fm(x))Tsubject to:xΩwhere Ω  Rn is the feasible region of the decision space and x = (x1, x2, …, xn)  Ω is the decision variable vector. The target function F(x) : X  Rm consists of m real-valued continuous objective functions f1(x), f2(x), …, fm(x) and Rm is the objective space.

Since no single solution can optimize all the objectives at the same time, the solution of a MOPs is a set of decision variable vectors rather than a unique solution. Let xA, xB  Ω be two decision vectors, xA is said to dominate xB, denoted by xA  xB, if fi(xA)  fi(xB) for all i  {1, …, m}, and there exists an j  {1, …, m } that satisfies fj(xA) < fj(xB). A decision vector x*  Ω is called the Pareto optimal solution or non-dominated solution if there is no other x  Ω that satisfies x  x*. The set of all the Pareto optimal solutions is called the Pareto set (PS) and the set of all the Pareto optimal objective vectors is the Pareto front (PF). Knowing that it is impossible to find the whole PS of continuous MOPs, we aim at finding a finite set of Pareto optimal vectors which are uniformly scattered along the true PF and are therefore highly representative of the entire PF.

With the advantage of producing a set of Pareto optimal solutions in a single run, evolutionary algorithms (EAs) have been found to be very successful in solving MOPs. Since the pioneer work of Schaffer [1], a number of multi-objective evolutionary algorithms (MOEAs) had been developed. In the early 1990s, the first generation of MOEAs which were characterized by the use of selection mechanisms based on Pareto ranking and fitness sharing to maintain diversity were proposed. The list of representative algorithms includes the multi-objective genetic algorithm (MOGA) [2], the niched Pareto genetic algorithm (NPGA) [3] and the non-dominated sorting genetic algorithm (NSGA) [4]. Since the end of the 1990s, the second generation of MOEAs using the elitism strategy had been presented. The major contributions includes the strength Pareto evolutionary algorithm (SPEA) [5] and its improved version SPEA2 [6], the Pareto archived evolution strategy (PAES) [7], the Pareto envelope based selection algorithm (PESA) [8] and its revised version PESAII [9], and the improved version of NSGA (NSGAII) [10]. The above mentioned MOEAs share more or less the same framework as that of NSGA-II, while in recent years some new frameworks have been developed. The MOEA/D [11] which combines the decomposition method in mathematics and the optimization paradigm in evolutionary computation is the most representative algorithm. It outperforms other MOEAs based on Pareto selection in solving MOPs with complicated PS shapes [12]. The indicator based framework [13] uses the performance indicators of a MOEA to guide the search for elite solutions of a MOP. The preference [14] and interactive [15] based framework uses the preference information from the decision maker to guide the search towards the PF region of interest. Many other nature inspired meta-heuristic search techniques have also been applied to solving MOPs, such as the simulated annealing algorithms [16], the particle swarm based algorithms [17], the ant colony based algorithms [18], the immune based algorithms [19], [20], [21] and the estimation of distribution algorithm [22], [23]. Moreover, MOEAs for complicated MOPs have also been extensively investigated which include MOEA for dynamic MOPs [24], constraint MOPs [25] and many objective optimization problems [26].

Artificial immune system (AIS) is a new type of computational system which is inspired by the theoretical immunology and observed immune functions, principles and models. In recent years, AIS has received a significant amount of interest from researchers and industrial sponsors. It has been used to solve various types of problems such as fault diagnosis, computer security, pattern recognition, scheduling and optimization [27]. In the field of immune optimization computing, more and more researches indicate that, comparing with EAs, artificial immune algorithms can maintain better population diversity and thus they do not easily fall in local optima. Recently, Coello proposed the multi-objective immune system algorithm (MISA) [28] based on the immune clonal selection principle. Jiao, Gong and Yang et al. proposed the multi-objective immune algorithms NNIA [19] and AHM [20]. AIS has drawn the attention of many researchers on MOPs solving, resulting in a lot of multi-objective artificial immune algorithms with improved performance. In this work, we present an immune inspired multi-objective optimization approach with a novel recombination operator which is enlightened by differential evolution.

Differential evolution (DE) has recently emerged as a simple and efficient algorithm for global optimization over continuous spaces [29]. Over the past decade, DE has been successfully combined with other optimization techniques to form hybrid DE algorithms. One type of hybrid DE algorithms introduce local search techniques into DE to improve its exploitation abilities [30], [31], [32]. The other type of hybrid DE algorithms combine DE with other global optimization algorithms like PSO [33], ant colony systems [34], and artificial immune systems [35]. At present, many variants of DE have been proposed to solve MOPs. A state-of-the-art literature survey indicates that DE has already proved itself as a promising candidate in the field of evolutionary multi-objective optimization (EMO) [36]. In many widely used MOPs, the PS in the decision space can be defined by piecewise linear functions. Due to the capability of DE based evolutionary operator which can exploit linear interdependencies between decision variables in a problem, it is suitable for solving MOPs. In this work, we design a novel DE inspired recombination operator to enhance the performance of an immune multi-objective optimization algorithm. In the new recombination operator, two types of searching strategies which find target vectors along current PS and towards wider regions respectively are designed according to the features of the continuous MOPs.

This paper provides an immune multi-objective optimization algorithm with differential evolution inspired recombination (IMADE). In IMADE, the fitness value of each non-dominated individual is assigned according to its crowding-distance [10]. Thus, IMADE pays more attention to the less-crowded regions of current PF in a single iteration. In IMADE, a novel DE inspired recombination is designed according to the features of the MOPs. In the newly designed recombination operator, two types of searching directions which explore diversity along current PS and move towards the true PS are determined according to the other two neighboring individuals in the current PS.

The remainder of this paper is organized as follows: Section 2 provides a brief description of immune inspired optimization framework, as well as some basic concepts used in this paper. Section 3 describes the main loop of IMADE, with a particularly detailed description and analysis of the proposed DE recombination operator. In Section 4, the computational complexity of IMADE will be analyzed. Section 5 investigates the experimental study used to assess our proposed approach and the efficiency of the specially designed DE recombination. In Section 6, concluding remarks are presented.

Section snippets

Immune inspired optimization approach

Optimization is one of the most important application fields of artificial immune systems. In the view of immune optimization computing, the process of the human immune system recognizing and responding to the antigens can be considered as a space searching procedure with the aim of finding a best-matching antibody in the antibody space. If we consider the optimization problem and its constraints as the antigen, and the candidate solution of the target problem as the antibody, the process of

Methodology

In this section, we will firstly introduce the basic idea and the general framework of the proposed algorithm IMADE. The new DE inspired recombination is then described and analyzed in the following section.

Computational complexity of IMADE

The computational complexity of IMADE is analyzed. In the field of evolutionary multi-objective optimization, researchers tend to analysis computational complexity by considering the number of objectives and the population size, because the running time of an evolutionary multi-objective optimization mainly depends on the selection operator which is tightly related to the above two factors. Assume that the number of objectives is m, the memory size is ms, the proliferation size is ps, the clone

Experimental study of IMADE

In this section, we will compare the proposed IMADE with two outstanding evolutionary multi-objective optimization approaches PESAII [9], NSGAII [10], an immune based multi-objective optimization algorithm NNIA [19] and an advanced differential evolution based multi-objective optimization algorithm OWMOSaDE [43], in solving 10 well-known multi-objective test instances. The simulation softwares of the compared approaches are developed by VC++6.0 and run on a personal computer with Inter Core2

Concluding remarks

Differential evolution is a simple yet efficient evolutionary algorithm for continuous optimization problems. In recent years, many improvements have been made by introducing differential evolution to multi-objective optimization algorithms. In this work, we have proposed a hybrid immune multi-objective optimization algorithm with a novel differential evolution based recombination (IMADE). In the proposed IMADE, the searching paradigm of immune clonal selection has been employed and a novel

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61303119 and 61373043, and the Fundamental Research Funds for the Central Universities under Grant Nos. JB140304.

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