自适应差分进化：增强多目标免疫算法的性能

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本文主要探讨了"自适应差分进化新型混合多目标免疫算法" (ADE-MOIA)，这是一种创新的多目标优化方法。在这个算法中，作者 Qiuzhen Lin、Qingling Zhu、Peizhi Huang、Jianyong Chen、Zhong Ming 和 Jianping Yu 将差分进化 (DE) 技术巧妙地融入到免疫算法 (MOIA) 中，旨在提升解决多目标优化问题 (MOPs) 的效率和鲁棒性。 DE 是一种强大的全局搜索算法，以其并行性和灵活性而著称，特别适合处理复杂的搜索空间。而在 MOIA 中，免疫系统的概念被应用于优化，模拟生物体的进化过程来寻找最优解。然而，单纯使用 DE 或 MOIA 可能存在局限性，因此作者提出了一种新颖的自适应 DE 操作，以增强算法的性能。这一操作的关键要素包括： 1. 自适应父代选择策略：在选择用于交叉和变异的个体时，算法不仅从当前进化群体中选取，还考虑了优势种群中的个体，这有助于引入更多元化的基因组合，提高搜索的效率。 2. 自适应参数控制：DE 的交叉率和比例因子不再是固定值，而是根据每个个体的进化进度和成功概率动态调整。这种自适应策略允许算法根据问题特性实时调整策略，从而更好地平衡收敛速度和整体多样性。作者通过实验对比 ADE-MOIA 与 NSGA-II、SPEA2、AbYSS、MOEA/D-DE、MIMO 和 D2MOPSO 等知名算法，在21个标准的多目标问题上进行了评估。结果显示，ADE-MOIA 在大多数情况下表现出显著的优势，显示出其在解决复杂多目标优化问题时的优越性。这一研究不仅提升了混合免疫算法的性能，也为其他多目标优化算法的设计提供了有价值的借鉴。总结来说，本文的核心贡献在于提出了一个自适应的混合算法框架，有效结合了 DE 的全局搜索能力和 MOIA 的进化策略，通过动态调整参数实现了对多种多目标优化问题的高效求解。这一成果对于优化领域的实践者来说，无疑为解决实际问题提供了强有力的工具。

generations, is satisﬁed, the non-dominated antibodies in external

archive are exported as the ﬁnal result.

When initialization, an initial population P ¼fx

; x

; :::; x

g is

created, where the decision variables of x

ði ¼ 1; 2; :::; NÞ are ran-

domly generated in Ω and the corresponding objective values are

evaluated. After performing the fast non-dominated sorting [16] in

P, the non-dominated and dominated individuals are obtained and

respectively preserved in the external archive (EXA) and domi-

nated archive (DA). The pseudo-code of initialization, named

Algorithm 1, is described in Fig. 2.

3.1. Cloning operator

It is assumed that the evolutionary population is P with size N

and the cloning population is P

¼fa

; a

; :::; a

g with size NC.At

ﬁrst, NC antibodies having the larger crowding-distance values are

selected from EXA. Afterward, the cloning is activated and the

evolutionary population will be

P ¼[

i ¼ 1

 a

gð6Þ

where q

stands for the number of clones corresponding to each

antibody a

ði ¼ 1; 2; :::; NCÞ, calculated by

¼ N 

CDða

j ¼ 1

CDða

&’

ð7Þ

where CDða

Þ is the ﬁtness value of antibody a

ði ¼ 1; 2; :::; NC Þ and

set as the crowding-distance value, computed by the following

equation:

CDða

Þ¼

j ¼ 1

ða

max  f

min

ð8Þ

where m is the number of objectives, f

max and f

min are

respectively the maximal and minimal values of the jth objective,

ða

Þ is the crowding distance of the jth objective for the

antibody a

ði ¼ 1; 2; :::; NC Þ and obtained by

It is noted that when the antibody locates in the boundary, its

crowding distance is 1 and the number of clones cannot be

obtained by Eq. (7). In this case, the crowding distance of the

boundary antibody is set as twice of the maximal crowding

distance except for the boundary solutions. The pseudo-code of

this operator named Algorithm 2 is shown in Fig. 3, where NC is

the size of clone population and the function CrowdingDistan-

ceAssignment(P) calculates the crowding distance value of each

antibody in population P.

3.2. Evolutionary operator

3.2.1. DE operator

Differential evolution has very strong global search capability

and shows pretty good convergence ability [27–34]. Thus, DE

operator is widely adopted in MOEAs [29–34]. In ADE-MOIA, a

novel ADE operator is presented, which is tailor-made to effec-

tively cooperate with MOIA. The weakness of MOIAs in population

diversity can be repaired by the strong global search capability of

DE. Our proposed ADE operator includes a suitable parent selec-

tion strategy and a novel adaptive parameter control approach.

This makes ADE operator become more effective for MOIAs. The

ADE operation is given as follows:

¼ x

þFðx

x

Þð10Þ

where F A 0ð ; 1:0



is a scaling factor. It is important to note that the

selected parent vectors used in Eq. (10) have great impact on the

optimization performance. The most popular DE operators include

DE/rand/1/bin, DE/best/1/bin, DE/current-to-best/1/bin, DE/rand/

2/bin, DE/best/2/bin [28], which are mostly adopted to solve SOPs

and especially effective for some SOPs with certain complex

features. When solving MOPs, these popular DE operators are

not perfectly ﬁt as all the non-dominated solutions are regarded as

the best candidate for MOPs. The selection of parent vectors in

MOPs is challenging. In our proposed ADE operator, in order to

promote the population diversity, three parent vectors picked

from the EXA and DA are aimed at providing a correct evolutionary

ADE operator

Update archive

Cloning operator

Initialization

Output result

Yes

Perturbance

Termination?

Fig. 1. The algorithmic ﬂowchart of ADE-MOIA.

Algorithm 1: Initialization

for i = 1 to N

randomly generate an individual

evaluate the objectives of

add

to the population P

end for

EXA = Find_nonDominated( P ); //find non-dominated solutions in P

DA = Find_Dominated ( P ); //find the dominated solutions in P

Fig. 2. The pseudo-code of initialization.

Algorithm 2: Cloning Operator

= EXA;

if ( |P

| > NC )

CrowdingDistanceAssignment(P

);

= Sort( P

);// sort P

according to the crowding distance in descending order

= SelectforClone( P

);//select the first NC antibodies in P

end if

for i = 1 to |P

calculate the number q

of clones for

according to Eq. (7)

clone q

individuals of

and add them to P

end for

Fig. 3. The pseudo-code of cloning operator.

ða

Þ¼

1; if ðf

ða

Þ¼¼f

min or f

ða

Þ¼¼f

maxÞ;

min ff

ða

Þf

ða

Þg; f

ða

Þ4 f

ða

Þ4 f

ða

Þðk; lA ½1; NÞ; otherwise

(

ð9Þ

Q. Lin et al. / Computers & Operations Research 62 (2015) 95–11198

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