不完全免疫算法简介AIMA–AIS学习笔记7

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多目标优化

An adaptive MOIAs with multiple DE

参考文献
An adaptive immune-inspired multi-objective algorithm with multiple differential evolution strategies,Information Sciences 430–431 (2018) 46–64

摘要

Most multi-objective immune algorithms (MOIAs) adopt clonal selection to speed up convergence, as this operator only clones the best individuals during the search process. However, this approach somehow deteriorates the population diversity, which may cause a MOIA to be trapped in a local optimum and could also lead to premature convergence when tackling some complicated multiobjective optimization problems (MOPs). In order to overcome this problem, an adaptive immune-inspired multi-objective algorithm (AIMA) is presented in this paper, in which multiple differential evolution (DE) strategies having distinct advantages are embedded into a conventional MOIA. Our proposed approach strengthens the exploration capabilities of a MOIA while also improving its population diversity. At each generation, based on the current search stage, an adaptive selection method is designed to choose an appropriate DE strategy for evolution. The core idea is to effectively combine the advantages of three DE strategies when solving different MOPs. A number of comparative experiments are conducted on the well-known and frequently- used WFG and DTLZ test problems. Our experimental results validate the superiority of our proposed AIMA, as it performs better than some state-of-the-art multi-objective opti- mization algorithms and some state-of-the-art MOIAs.

针对MOEA有很多种改进方式

Due to the population-based nature of evolutionary algorithms (EAs), they are very suitable for tackling MOPs since they can process a set of solutions in a single run. During the recent decades, a number of multi-objective EAs (MOEAs) have been designed, showing a very promising performance on tackling different MOPs. There are three well-known representa- tives of state-of-the-art MOEAs, i.e. , NSGA-II [11] , SPEA2 [53] , and MOEA/D [49] . Regarding NSGA-II [11] , it incorporates a fast nondominated sorting approach to direct the search, while a crowding-distance metric is used to maintain the population’s diversity. On SPEA2 [53] , a nearest neighbor density estimation technique is proposed to maintain the population’s diver- sity, combined with a fine-grained fitness assignment strategy that is used to guide the search. For MOEA/D [49] , a MOP is decomposed into a set of SOPs and then these SOPs are solved on a cooperative manner using evolutionary search. These state-of-the-art MOEAs have inspired many enhanced variants. For example, regarding NSGA-II, a novel parent inheritance operator was embedded and several jumping gene adaptations were used in [32] to speed up convergence towards the global PF, while a reference point based approach was introduced in [12] to maintain the population’s diversity when tack- ling many-objective optimization problems ( i.e. , MOPs having more than three objectives); for SPEA2, a shift-based density estimation (SDE) strategy [26] was presented to enhance its performance on tackling many-objective optimization prob- lems; with respect to MOEA/D, a dynamic resource allocation (DRA) strategy was introduced in [50] to dynamically assign the computational resources based on the difficulties of sub-problems, and an economic stable matching model (STM) was designed in [25] to guarantee the balanced match of sub-problems and solutions by mutual preferences. A detailed review of MOEAs can be found in [33] .

其中加入DE是一个很不错的方式

Especially, some of MOEAs were enhanced based on the use of differential evolution (DE), since DE shows excellent search capabilities. The experiments conducted by [4] and [41] showed that DE can significantly enhance the performance of MOEAs, as the DE-based variants of three state-of-the-art MOEAs ( i.e. , NSGA-II, SPEA2 and IBEA) significantly outperformed the original ones. In MOEA/D-DE [23] and CMODE [43] , a specific DE operator was used to substi- tute the original evolutionary operators of MOEA/D [49] and CMPSO [48] , giving rise to a better optimization performance; Moreover, in ADEMO/D [42] , MOEA/D-FRRMAB [24] , and MOEA/D-CDE [31] , multiple DE operators were further combined to enhance their performance. These promising results have evidenced the advantages of incorporating single or multiple DE operators into a MOEA.

其中有文献在MOIA中加入DE取得了不错的结果

On the other hand, multi-objective immune algorithms (MOIAs) are designed to mimic the process of clonal selection [3,6,19,37] , as inspired from the biologic immune system. The nondominated neighbor-based immune algorithm (NNIA) [19] may be the first real-coded MOIA using the clonal selection approach. Since the report of NNIA, a number of other MOIAs have been designed and enhanced under its framework [6,22,29] . Among these MOIAs, clonal selection is em- ployed to pick out a few of less-crowded nondominated solutions, which are then proportionally cloned according to their crowding-distance values [11] . Then, the clones undergo the heuristic search operations, such as recombination and muta- tion. By this way, the less-crowded search area will be assigned with more clones for exploration. Note that the boundary area is considered as the sparsest area and, therefore, it will be explored by more clones. Compared to the selection operator in NSGA-II [11] and most of other MOEAs [12,33,43] , clonal selection enables MOIAs to allocate more search effort s to the boundary and less-crowded areas, which helps to improve the convergence speed and tries to extend the population’s diver- sity. However, when dealing with some complicated MOPs, MOIAs may easily fall into local optimum and suffer from prema- ture convergence or stagnation due to the lack of population diversity [22,38] , as only a few of nondominated solutions are selected for cloning, especially at the early stages of the search. To overcome this limitation, some MOIAs [27,28,30,34] have been recently proposed to embed the DE operators. These embedded DE operators have been often used to replace or coop- erate with the simulated binary crossover (SBX) operator [14] , as the DE operators normally show a better search capability than SBX [28] . Therefore, embedding them can help to enhance the population diversity of MOIAs

我们做一个加入多个DE算子的尝试

Therefore, in this paper, an Adaptive Immune-inspired Multi-objective Algorithm, called AIMA, is proposed in this paper. Three DE strategies with different parameters settings are embedded into a state-of-the-art MOIA ( i.e. , NNIA [19] ) and an adaptive DE strategy selection approach is designed to automatically run an appropriate DE strategy at each generation based on the current evolutionary stage. These three DE strategies can provide different search properties. Thus, they can significantly enhance the search capability and population diversity of NNIA when appropriately selected using the adaptive DE strategy. When solving the well-known and frequently- used test MOPs (WFG [20] and DTLZ [13] ) with various features, AIMA shows evident advantages over five state-of-the-art multi-objective optimization algorithms ( i.e. , NSGA-II [11] , SPEA2 [53] , MOEA/D [49] , SMS-EMOA [2] , and CMPSO [48] ) and their DE-based variants [23,43] , and four competitive MOIAs ( i.e. , NNIA [19] , IMADE [34] , DMMO [27] , and HEIA [28] ).

A brief introduction to MOIAs

The first MOIA was designed in [47] to solve a multi-criterion design problem, in which the concept of antibody-antigen affinity is used to modify the fitness assignment mechanism of a standard genetic algorithm. After that, a variety of MOIAs were designed with the aim of enhancing performance. Based on the special features inspired by the immune system, most MOIAs can be classified into three main categories.

clonal selection approach

The first kind of MOIAs is designed based on the clonal selection approach [3] , which applies the cloning operator to produce copies of antibodies having the highest affinity values. In the multi-objective immune system algorithm (MISA) [8] , the antibodies with high affinities were cloned to have multiple copies and an adaptive grid was used to maintain the population’s diversity. In the immune dominance clonal multi-objective algorithm (IDCMA) [21] , the antibody-antibody affinity was used to reflect the similarity among antibodies, which helps to guide the application of the cloning operator on an effective search region ( i.e. , the least-crowded region). This approach was modified to implement a real-coded MOIA in NNIA [19] . In the hybrid immune multi-objective algorithm (HIMO) [6] , a hybrid operator combining Gaussian mutation and polynomial-based mutation was designed, which was further enhanced by using an adaptive mutation operator in the micro-population immune multi-objective algorithm (MIMO) [29] .
There is a second class of MOIAs which adopts immune network theory to evolve the population and to maintain the population’s diversity. In the vector artificial immune system (VAIS) [16] , two evolutionary loops were performed, in which the inner loop is used to exploit the search space, while the outer one using the suppression mechanism is adopted to lower the redundancy among similar antibodies. In the weight-based MOIA [17] , a random weighted sum method was used as a fitness assignment scheme and a new truncation approach was presented to eliminate similar individuals.
The last kind of MOIAs combines an immune system with another heuristic, by embedding operators of another heuris- tic into MOIAs. In the immune-inspired Pareto archived evolution strategy [9] , two hypermutation operators were integrated to solve a MOP extracted from the problems of ab initio protein structure prediction. In the evolutionary artificial immune algorithm [40] , the global search capabilities of EAs were combined with the learning capabilities of artificial immune sys- tems. In the immunity-based hybrid evolutionary algorithm [46] , a sorting scheme featuring uniform crossover, multi-point mutation, and crowding distance sorting were hybridized to solve both unconstrained and constrained MOPs.
some paper new In recent years, some new MOIAs with competitive performance have been designed. For example, a novel MOIA was introduced in [22] with two interesting methods. One is to design a resource allocation model to speed up convergence, while the other is to use a double-sphere crowding distance measure to improve the population’s diversity. In [34] , the im- mune multi-objective optimization algorithm with differential evolution inspired recombination (IMADE) was reported. This approach presents a novel recombination operator, combining a newly designed DE operator and simulated binary crossover (SBX) [14] . Therefore, it provides two different search patterns during the search process. A double-module immune algo- rithm for MOPs (DMMO) [27] was presented with two evolutionary modules, with the aim of simultaneously enhancing the convergence speed and the population’s diversity. The first module is used to optimize each objective independently using DE in each sub-population, while the second module follows the traditional procedures of MOIAs to optimize multiple con- flicting objectives. In [30] , a novel hybrid multi-objective immune algorithm with adaptive differential evolution (ADE-MOIA) was designed. This approach introduces an adaptive DE operator to further enhance the robustness of MOIAs in solving various kinds of MOPs. More recently, the hybrid evolutionary immune algorithm for MOPs (HEIA) [28] was reported, providing a novel hybrid evolutionary framework for MOIAs. This approach divides the cloned individuals into several sub-populations, which are then separately evolved by using different evolutionary strategies. An implementation of this framework with two evolutionary strategies (SBX and DE) is shown in HEIA.

Clonal selection operator in NNIA

To illustrate the way in which the clonal selection operator works, here a well-known state-of-the-art MOIA ( i.e. , NNIA [19] ) is introduced. This approach has inspired the design of many other MOIAs [6,22,29,30,34] . Its main contribution is to propose a nondominated neighbor-based selection and a crowding-distance-based proportional cloning mechanism. These two approaches actually constitute the clonal selection operator in NNIA. To show the way in which this operator works, in Fig. 1 , we show the population evolution in a single generation at time t for NNIA.
According to the above process of clonal selection, it can be easily observed that the clonal selection operator in NNIA puts more clones to search the less-crowded and boundary areas, thus it shows very promising convergence speed, as val- idated in [19,28] . However, according to the No-Free-Lunch theorem [45] , as we produce more clones of the best solutions,the higher will be the loss of population diversity [38] . As experimentally validated in [38] , when the nondominated so- lutions are few (this case often happens at the early stage of the search), the population diversity in NNIA is extremely poor and it is very easy to get trapped into a local optimum when solving some complicated MOPs with many local PFs ( e.g. , DTLZ1, DTLZ3 and WFG1) or with disconnected PFs ( e.g. , DTLZ7 and WFG2). To overcome this problem, some MOIAs [27,28,30,34] have tried to embed the search capabilities of DE. Although these MOIAs were shown to obtain promising results, they only adopted one single DE strategy, which provides limited search capabilities and may not perform well in tackling various types of MOPs with different features. Therefore, this paper proposes to embed three DE strategies with dif- ferent parameter settings into a MOIA and then adaptively select an appropriate DE strategy for running at each generation, aiming to enhance the search capabilities of MOIAs during the different search stages, as introduced in Section 3 .