JADE：带外部归档的自适应差分进化算法

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"这篇论文提出了一种新的差分进化（DE）算法——JADE（Adaptive Differential Evolution with Optional External Archive），旨在通过引入一种新的突变策略‘DE/current-to-pbest’以及可选的外部归档功能，以自适应的方式更新控制参数来提升优化性能。" 差分进化算法（Differential Evolution，DE）是一种全局优化算法，主要用于解决连续函数的最优化问题。它基于群体的迭代过程，通过变异、交叉和选择操作来寻找最佳解。在经典的DE算法中，变异操作通常采用“DE/current-to-best”策略，即当前个体与最好个体之间的差异加到当前个体上。 JADE算法对此进行了扩展，提出了“DE/current-to-pbest”策略，这是一种对“DE/current-to-best”的一般化形式。在“DE/current-to-pbest”中，不仅考虑了当前个体和最好个体之间的差异，还可能包含历史最优个体的信息，这样可以增加种群的多样性，避免过早收敛，同时也能改善算法的收敛性能。论文中引入的可选外部归档机制是另一个创新点。外部归档用于存储历史最优解，这些历史数据可以提供优化过程中的进步方向信息。这种机制使得算法能够利用过去的信息，帮助算法跳出局部最优，从而提高全局搜索能力。 JADE算法的另一个关键特征是其参数自适应更新机制。传统DE算法通常需要用户预设控制参数，如种群规模、变异因子和交叉概率等，而JADE算法能自动调整这些参数至合适的值，无需用户具备对问题特性和参数设置之间关系的先验知识。这增强了算法的鲁棒性，使其能更好地适应各种复杂优化问题。通过模拟实验，JADE算法表现出了优于或至少与其它先进DE算法相当的性能。这些结果验证了JADE在优化效率和适应性方面的优势，使得它成为解决高维度和复杂优化问题的一个有力工具。JADE算法为差分进化算法的改进提供了一个新的视角，强调了动态调整和利用历史信息对于优化性能的重要性。

ZHANG AND SANDERSON: JADE: ADAPTIVE DIFFERENTIAL EVOLUTION WITH OPTIONAL EXTERNAL ARCHIVE 947

violated bounds and the corresponding components of the

parent individual [2], i.e.,

j,i,g

= (x

low

+ x

j,i,g

)/2, if v

j,i,g

< x

low

j,i,g

= (x

+ x

j,i,g

)/2, if v

j,i,g

> x

where v

j,i,g

and x

j,i,g

denote the jth components of the

mutation vector v

i,g

and the parent vector x

i,g

at generation

g, respectively. This method works well especially when the

optimal solution is located near or on the boundary.

Crossover: After mutation, a binomial crossover op-

eration forms the ﬁnal trial/offspring vector u

i,g

1,i,g

, u

2,i,g

,...,u

D,i,g

)

j,i,g



j,i,g

, if rand(0,1) ≤ CR

or j = j

rand

j,i,g

, otherwise

(4)

where rand(a, b) is a uniform random number on the interval

[a, b] and independently generated for each j and each i,

rand

= randint(1, D) is an integer randomly chosen from

1toD and newly generated for each i, and the crossover

probability CR

∈ [0, 1] roughly corresponds to the average

fraction of vector components that are inherited from the

mutation vector. In classic DE, CR

= CR is a ﬁxed parameter

used to generate all trial vectors at all generations, while in

many adaptive DE algorithms each individual i is associated

with its own crossover probability CR

Selection: The selection operation selects the better one

from the parent vector x

i,g

and the trial vector u

i,g

accord-

ing to their ﬁtness values f (·). For example, if we have a

minimization problem, the selected vector is given by

i,g+1



i,g

, if f (u

i,g

)< f (x

i,g

)

i,g

, otherwise

(5)

and is used as a parent vector in the next generation.

The operation in (5) is called a successful update if the trial

vector u

i,g

is better than the parent x

i,g

, i.e., the improvement

or evolution progress 

i,g

= f (x

i,g

) − f (u

i,g

) is positive.

Accordingly, the control parameters F

and CR

used to

generate u

i,g

are called a successful mutation factor and a

successful crossover probability, respectively.

The above one-to-one selection procedure is generally kept

ﬁxed in different DE algorithms, while the crossover may

have variants other than the binomial operation in (4). Thus, a

classic differential evolution algorithm is historically named,

for example, DE/rand/1/bin, connoting its DE/rand/1 mutation

strategy and binomial crossover operation.

III. A

DAPTIVE DE ALGORITHMS

This section brieﬂy reviews some recent adaptive and

self-adaptive DE algorithms that dynamically update control

parameters as the evolutionary search proceeds. It provides

a convenient way to compare these parameter adaptation

mechanisms with the one proposed in JADE later.

A. DESAP

In [10], an algorithm DESAP is proposed to not only

dynamically adapt mutation and crossover parameters η and

δ (which is usually the case in other adaptive or self-adaptive

algorithms) but also the population size π . The base strategy

of DESAP is slightly different from the classic DE/rand/1/bin

and rather similar to the scheme introduced in [9]—while δ

and π have the same meaning as CR and NP, respectively, η

refers to the probability of applying an additional normally

distributed mutation after crossover. The ordinary mutation

factor F is indeed kept ﬁxed in DESAP.

In DESAP, each individual i is associated with its own

control parameters δ

, η

,andπ

. These parameters undergo

crossover and mutation, in a way similar to the corresponding

vector x

. The new values of these control parameters survive

together with u

if f (u

)< f (x

). In spite of its simple reason-

ing, DESAP performance is not satisfactory. It outperforms the

conventional DE only in one of the ﬁve De Jong’s test prob-

lems, while other results are very similar. Indeed, as the author

states, DESAP is mainly used to demonstrate a possibility of

further reducing control parameters by adaptively updating the

population size as well as other control parameters.

B. FADE

The fuzzy adaptive differential evolution (FADE), which

was introduced by Liu and Lampinen [11], is a new variant

of DE that uses fuzzy logic controllers to adapt the con-

trol parameters F

and CR

for the mutation and crossover

operations. A similar method has also been independently

proposed for multiobjective DE optimization [12]. Similar to

many other adaptive DE algorithms (except DESAP [10]), the

population size is assumed to be tuned in advance and kept

ﬁxed throughout the evolution process of FADE. The fuzzy-

logic controlled approach is tested with a set of 10 standard

benchmark functions and shows better results than the classic

DE when problem dimension is high.

C. SaDE

SaDE [13] is proposed by Qin and Suganthan to simulta-

neously implement two mutation strategies “DE/rand/1” and

“DE/current-to-best/1.” It adapts the probability of generating

offspring by either strategy based on their success ratios in the

past 50 generations. It is believed that this adaptation proce-

dure can gradually evolve the most suitable mutation strategy

at different learning stages for the problem under consideration

This is similar to the scheme proposed in [27], [28], where

competing heuristics (including several DE variants, simplex

methods, and evolution strategies) are simultaneously adopted

and the probabilities of generating offspring by any of these

heuristics are adapted dynamically.

In SaDE, the mutation factors are independently generated

at each generation according to a normal distribution with

mean 0.5, standard deviation 0.3, and truncated to the interval

(0, 2]. It states that this scheme can keep both local (with small

values) and global (with large F

values) search ability

to generate potentially good mutation vectors throughout the

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