How to maintain the population diversity is an important issue in designing a multiobjective evolutionary algorithm. This paper presents an enhanced nondominated neighbor-based immune algorithm in which a multipopulation coevolutionary strategy is introduced for improving the population diversity. In the proposed algorithm, subpopulations evolve independently; thus the unique characteristics of each subpopulation can be effectively maintained, and the diversity of the entire population is effectively increased. Besides, the dynamic information of multiple subpopulations is obtained with the help of the designed cooperation operator which reflects a mutually beneficial relationship among subpopulations. Subpopulations gain the opportunity to exchange information, thereby expanding the search range of the entire population. Subpopulations make use of the reference experience from each other, thereby improving the efficiency of evolutionary search. Compared with several state-of-the-art multiobjective evolutionary algorithms on well-known and frequently used multiobjective and many-objective problems, the proposed algorithm achieves comparable results in terms of convergence, diversity metrics, and running time on most test problems.

Optimization problems widely exist in real life, especially in engineering applications [

The first study on multiobjective evolutionary algorithm (MOEA) is probably the vector evaluated genetic algorithm (VEGA) [

The concept of artificial immune systems (AIS) was first put forward in 1996. Since then, AIS have stepped into a high-speed development period and become one of the hot topics in the field of artificial intelligence. AIS that get the inspiration from biological immune systems attempt to develop computational tools for solving science and engineering problems. Some AIS-based multiobjective optimization algorithms have been proposed [

An enhanced version of nondominated neighbor-based immune algorithm with a multipopulation coevolutionary strategy is proposed for improving the population diversity. Subpopulations employ evolutionary operations independently; thus the unique characteristics of each subpopulation can be effectively maintained. During evolutionary search, information exchanges among subpopulations thus expanding the search range of the entire population. As a matter of fact, most of the evolutionary algorithms employ regular operations throughout the whole evolutionary process, and few of them take advantage of online discovered information. The adaptive operator which dynamically applies evolution operations to subpopulations based on the online discovered information is designed. Therefore, evolutionary search becomes more directional and purposeful and the unnecessary waste of computational cost is reduced.

The remainder of this paper is organized as follows. The problem statement is described in Section

The mathematical description of multiobjective optimization problems can be expressed as follows [

For a certain decision variable vector

For any two feasible solutions, if and only if they satisfy condition (

For a certain feasible solution

The corresponding image of the Pareto-optimal set in the objective space is called the Pareto-optimal front, which can be denoted as

In solving MOPs, it is expected that the set of nondominated solutions obtained by the proposed algorithm can well approximate the true Pareto-optimal front and the diversity of the solutions can be maximized.

Many new-type evolutionary methods have been introduced into the area of MOEAs. Immune-based algorithm is one of these late-model methods. Artificial immune systems (AIS) get the inspiration from biological immune systems. They have learnable, parallel, and distributed characteristics, therefore possessing an efficient information processing ability. AIS-based algorithms have attracted a lot of attention and have been applied to many complex MOPs, including constrained nonlinear MOPs and dynamic MOPs [

In NNIA, a nondominated neighbor-based selection and a crowding-distance-based proportional cloning were proposed. The fitness of a nondominated individual is assigned according to its crowding distance. The individual with greater crowding distance is reproduced more times and then less-crowded regions will have more chances to be searched, which improves the search ability of NNIA on less-crowded regions. Besides, only a minority of nondominated individuals will be selected to form an active population, and then a series of evolutionary operations are applied to this active population. Therefore, NNIA evolves very fast by performing evolutionary operations on a small-scale active population. The specific framework of population evolution in a single generation at time

Population evolution of NNIA.

From Figure

The situation of falling into local optimum.

Test problems | ZDT2 | ZDT4 |
---|---|---|

Times of falling into local optimum | 12 times | 10 times |

The number of nondominated solutions | 1 | 1 |

In this paper, we present an enhanced multipopulation coevolutionary strategy for nondominated neighbor-based immune algorithm, called CONNIA. Different from the traditional evolution, co-evolution recognizes the simultaneous existence of competition and cooperation among populations, which provides a theoretical basis for maintaining the population diversity.

When it comes to the adaptive operator in the field of MOEAs, it mainly refers to adaptively tuning some parameters, such as population size, crossover probability, and mutation probability. However, the adjustment of evolutionary strategy based on evolutionary conditions is seldom involved. The major contribution of the designed adaptive operator is that each subpopulation adaptively selects corresponding operators during the evolution, which makes the evolution become more purposeful and directional. Therefore, the need for unnecessary computing resource existing in random search is avoided effectively.

After performing a series of evolutionary operations on each subpopulation, a way for measuring the evolutionary condition is to identify the nondominated solutions of each subpopulation. The set coverage metric is employed for measuring the relationship between two subpopulations [

If the difference of the set coverage metric between subpopulations is not obvious, a local search operator would be employed. Two subpopulations perform independently evolutionary operations and search within different solution space for maintaining the diversity of the entire population. Meanwhile, some appropriate perturbations are applied around the obtained nondominated solutions for seeking the possible better solutions and reducing the probability of getting into local optimum.

If the difference of the coverage metric between subpopulations is obvious, a cooperation operator would be employed. The information exchanges among subpopulations, which reflects a mutually beneficial relationship between two subpopulations. The subpopulation with lower value of the set coverage metric could make use of the reference experience from another subpopulation to improve its own evolution. Two subpopulations make progress together by means of cooperation to ultimately complete the evolutionary task.

We get the inspiration from traditional differential evolution (DE) operator [

The designed local search operator and cooperation operator reflect a mutually beneficial relationship between subpopulations. Both operators transmitting information among antibodies within the same generation are combined with traditional evolutionary operators such as crossover and mutation, for transmitting information effectively.

The details of the proposed algorithm are described in this part. To be specific, the following parts are designed.

Generate two initial subpopulations

The nondominated antibodies of the two subpopulations

If the terminal condition is satisfied, export

Select the individuals which have more contributions to the population diversity from

Two clone populations

Perform recombination and mutation on

If the condition of information exchange is satisfied, perform cooperation operator between

Get subpopulations

In the proposed algorithm, the crowding distance [

The density around a dominant antibody is estimated by calculating its crowding distance. The larger the crowding distance of a dominant antibody is, the sparser the distribution around it will be, which also means that the contribution of this antibody to the population diversity is relatively greater. When it is required to delete some solutions, the antibody with small crowding distance will be deleted firstly. The traditional way of solution pruning is to calculate the crowding distance of all solutions only once, and then some solutions are deleted based on such one-shot result. However, such mechanism is unreasonable sometimes.

After calculating the crowding distance of all points shown in Figure

The static method of solution pruning.

Assume that the maximum size of the dominant population is

The time complexity for identifying nondominated individuals in the population is

Therefore the worst total time complexity is:

Owing to the fact that the operational rule of the symbol “

The time complexity for CONNIA in a single generation with information exchange can be calculated as follows.

The time complexity for identifying nondominated individuals in the population is

So the worst total time complexity is:

Owing to the fact that the operational rule of the symbol “

In real applications, the key factor to decide whether a technique can be applied is the running time. The further research on the practical running time of the proposed algorithm will be presented in Section

In this section, we compare CONNIA with three state-of-the-art MOEAs, including NNIA, NSGA-II, and SPEA2, on benchmark MOPs. Besides, some extensional problems based on the benchmark MOPs are also tested. It is well known that the parameter setting has significant impact on MOEAs. Therefore, the parameter setting of the four algorithms is consistent with the original references and has some adjustments appropriately. For SPEA2, the size of the population is 100; the size of an external population is 100. For NSGA-II, the size of the population is 100. For NNIA, the maximum size of the dominant population is 100; the maximum size of an active population is 20. For CONNIA, the maximum sizes of the two dominant subpopulations are both 50, and the maximum sizes of the two active subpopulations are both 10. A given number of function evaluations are used as the stopping criteria. We obtain statistical experimental results by running the four algorithms 30 times independently. To simplify the expression, Arabic numerals 1, 2, 3, and 4 are used to denote CONNIA, NNIA, NSGA-II, and SPEA2.

To evaluate various performances of the compared algorithms, some numerical metrics are adopted, including generation distance [

To verify the versatility of the proposed algorithm, five ZDT [

The related problems based on ZDT2 are described as follows. When

The shape of the Pareto-optimal front changes with the value of

The true Pareto-optimal fronts of the related problems based on ZDT2 and ZDT4, respectively.

Similar to the related problems based on ZDT2, the related problems based on ZDT4 are described as follows. When

The shape of the Pareto-optimal front changes with the value of

The influence of the threshold is discussed in this part. Considering the representative of multi-objective problems with two-objectives and three-objectives, respectively, ZDT4 and DTLZ3 are selected for parameter analysis. Figure

Mean values of GD and spacing versus the introduced parameter in solving DTLZ3 and ZDT4 by the proposed algorithm.

The cooperation operator reflects a mutually beneficial relationship between two subpopulations. By applying the cooperation operator, two subpopulations gain the opportunity to exchange information and expand the search range of the entire population. The subpopulation could make use of the reference experience from each other to improve its own evolution. This directed cooperation operator provides good evolutionary paths towards antibodies, thereby making antibodies evolve faster.

In this part, the effectiveness of information exchange among subpopulations is discussed. The proposed algorithm without information exchange is denoted by

(a) The error bar of HV of the nondominated antibodies in final population with different number of function evaluations by

Figure

The approximate Pareto-optimal fronts obtained by CONNIA, NNIA, NSGA-II, and SPEA2 in solving the 9 test problems.

Statistical values of convergence obtained by CONNI, NNIA, NSGA-II, and SPEA2 in solving the 10 test problems.

Statistical values of spacing obtained by CONNI, NNIA, NSGA-II, and SPEA2 in solving the 10 test problems.

Statistical values of maximum spread obtained by CONNI, NNIA, NSGA-II, and SPEA2 in solving the 10 test problems.

Statistical values of HV obtained by CONNI, NNIA, NSGA-II, and SPEA2 in solving the 10 test problems.

Figure ^{−3} in almost all the 30 independent runs by four algorithms on five ZDT problems. The box plots obtained by NNIA on ZDT2 and ZDT4 are quite broad which indicates that the stability of NNIA in solving these problems is quite poor. However, CONNIA is more robust than NNIA on ZDT2 and ZDT4, owing to the multipopulation coevolutionary strategy which plays an important role in maintaining the population diversity. In general, except for the appearance of local optimum when NNIA deals with ZDT2 and ZDT4, the differences among four algorithms on five ZDT problems are relatively small. Hereinto, CONNIA obtains the smallest values of convergence on ZDT3, ZDT4, and ZDT6. It has been pointed out that NSGA-II and SPEA2 could not completely converge onto the true Pareto-optimal fronts in a limited number of function evaluations on DTLZ3 which has some local Pareto-optimal fronts [

Figure

Figure

The comparison of CONNIA and NNIA on some difficult problems (DTLZ1 and DTLZ3) and some extreme problems (ZDT21, ZDT22, ZDT41, ZDT42, and ZDT43) is carried out in this part. Figure

The approximate Pareto-optimal fronts obtained by CONNIA and NNIA in solving the extensional problems.

Figure

Box plots of the coverage of two sets by CONNIA and NNIA in solving 10 test problems.

In the field of MOEAs, the number of function evaluations is commonly used as the stopping criteria. It is difficult to set the accurate stopping criteria for an MOEA on different problems, while uniform stopping criteria which are applied to different problems always provide a plethora of information [

Figure

Mean value of convergence with different function evaluations by the four algorithms in solving the 10 test problems.

In this section, the performance of four algorithms on many-objective problems is investigated. Multiobjective problems with more than three objectives are defined as many-objective problems. The test problems are the extensional problems of DTLZ1 and DTLZ2 with 4 to 7 objectives and are named DTLZ14–DTLZ17 and DTLZ24–DTLZ27, respectively. Due to the fact that the number of nondominated solutions dramatically enlarges with the number of objectives increasing, many MOEAs have difficulty in converging onto the true Pareto-optimal front with a limited number of function evaluations. Therefore, the size of population and the number of function evaluations are doubled as those in Section

Figure

Statistical values of convergence obtained by the four algorithms in solving many-objective problems.

Statistical values of spacing obtained by the four algorithms in solving many-objective problems.

Statistical values of maximum spread obtained by the four algorithms in solving many-objective problems.

The convergence metric can be only used under the condition of which knowledge of the true Pareto-optimal fronts is available, which is unsuitable for many-objective problems. Hence, the metric of the coverage of two sets is employed to measure the dominant relationship between solutions obtained by different algorithms. Figures

Statistical values of the coverage of two sets obtained by CONNIA and NNIA in solving many-objective problems.

Statistical values of the coverage of two sets obtained by CONNIA and NSGA-II in solving many-objective problems.

Statistical values of the coverage of two sets obtained by CONNIA and SPEA2 in solving many-objective problems.

Figure

Mean value of running time by four algorithms on the extensional problems of DTLZ1 and DTLZ2 with 3 to 7 objectives, respectively.

To the best of our knowledge, slow convergence rate is a ubiquitous problem in MOEAs. AIS have the learnable, parallel, and distributed characteristics and possess an efficient information processing ability. AIS-based algorithms have already been widely used for dealing with MOPs, in which NNIA obtains a fast convergence rate solving such knotty problem in MOEAs. However, the population diversity can not be well maintained in NNIA, which leads the solutions obtained by NNIA to be trapped into local optimum on some difficult problems. Co-evolution is a high-level evolutionary method, which confirms that all the populations are beneficial mutually, thus providing a theoretical basis for maintaining diversity. In this paper, a multipopulation coevolutionary strategy is designed. Subpopulations implement evolutionary operation independently; thus the diversity of the entire population can be well maintained. The information exchange among subpopulations is available, thereby expanding the search range of the entire population and improving the efficiency of evolutionary search.

In the field of MOEAs, when it comes to adaptive algorithms, most of them adaptively adjust some parameters, such as population size, crossover probability, and mutation probability. However, an adaptive algorithm with online-decision strategy is seldom involved. Based on this idea, an adaptive strategy is designed in the proposed algorithm. Subpopulations adopt corresponding operations according to the conditions of themselves which ensures that evolutionary search is not random or blind.

In dealing with many-objective problems, the rapid increase of nondominated solutions requires a large size of population or a large number of function evaluations. However, in many MOEAs, the size of population is constant. No matter how difficult the problem is, the size of population is the same. According to the characteristics of CONNIA, it is more reasonable to adaptively adjust the number of subpopulations according to the difficulty of the problem. We can imagine that it is more reasonable to employ more subpopulations together to cooperatively overcome the difficulty in solving many-objective problems.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant nos. 61273317, 61202176, and 61203303), the National Top Youth Talents Program of China, the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20130203110011), and the Fundamental Research Fund for the Central Universities (Grant nos. K50510020001 and K5051202053).