免疫记忆克隆算法在数值与组合优化中的应用

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"本文介绍了一种名为免疫记忆克隆算法（IMCA）的新颖人工免疫算法，该算法受到克隆选择理论和免疫网络模型的启发，适用于数值和组合优化问题。IMCA包括两种版本：自适应免疫记忆克隆算法（AIMCA）和免疫记忆克隆策略（IMCS）。算法的核心是克隆选择操作符，它能够结合全局搜索和局部搜索，提高搜索效率。此外，IMCA根据抗体-抗体和抗体-抗原的亲和力来自适应地调整存储单元的大小和抗体群体，有助于保持种群多样性并防止过早收敛。在实验部分，IMCA被应用于18个不同维度的多维函数以及旅行商问题和背包问题（KPs）等组合优化问题，显示出了较高的收敛速度和优秀的性能。理论分析表明，IMCA具有概率1的收敛性。该研究由高等教育出版社和施普林格出版社出版。" 本文详细探讨了人工免疫系统在优化问题中的应用，特别是提出了一种新的优化算法——免疫记忆克隆算法（IMCA）。IMCA借鉴了生物免疫系统的机制，特别是克隆选择理论，以解决数值优化和组合优化问题。在IMCA中，克隆操作符是一个关键元素，它允许算法在全局和局部搜索之间进行有效的平衡，以找到最优解。算法的两个变体——AIMCA和IMCS，分别通过动态调整抗体的突变率和存储单元大小，以及同时演化抗体种群和存储单元来提升性能。 AIMCA的创新之处在于，每个抗体的突变率和存储单元大小是动态调整的，这使得算法能够适应不断变化的搜索环境，提高搜索效率。另一方面，IMCS则关注抗体种群和记忆单元的整体进化，确保搜索过程的全面性和深度。在实验验证阶段，IMCA被应用于一系列多维函数优化问题，覆盖了广泛的维度范围，以及经典的组合优化问题如旅行商问题和背包问题。实验结果显示，IMCA在收敛速度上表现出色，而且能够在保持种群多样性的同时有效防止过早收敛，这对于避免陷入局部最优至关重要。此外，IMCA的理论分析表明，算法具有概率1的收敛性，这意味着在理论上，IMCA有保证地能找到问题的最优解或接近最优解的解决方案。这一特性使得IMCA在实际应用中更具吸引力。 IMCA是一种高效且适应性强的优化算法，它的设计灵感来源于生物学的免疫系统，通过模拟自然免疫机制来解决复杂优化问题。其在实验中的优秀表现和理论上的收敛性保证，证明了IMCA在数值和组合优化领域的潜在价值。

offprint

¼ exp

j¼1

, (2)

it is obvious that the smaller the Ab-Ab afﬁnity is, the

smaller l

is. Generally, before employed in the clonal

operator, l

is normalized to the interval [0, 1], and if

¼ 0, then l

¼ 1. l

can be used to adjust the scale of

clones and reduce the possibility of producing closely

related antibodies.

Based on the clonal selection theory and the immune

network mechanism, a new artiﬁcial intelligent operator is

proposed. The operator includes a polyclonal operator and

monoclonal operator, which are deﬁned as follows.

5.2 Monoclonal operator

Monoclonal operator is a compositive operator, which

integrates clone, clonal mutation, and clonal selection

together; it can be deﬁned as follows

Clone T

: I

! I

, namely

ðkÞ¼T

ðAðkÞÞ ¼ ½T

ða

ðkÞÞ,T

ða

ðkÞÞ,:::,T

ða

ðkÞÞ

¼ a

ðkÞ,:::,a

ðkÞ

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

, a

ðkÞ,:::a

ðkÞ,

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

:::, a

ðkÞ,:::a

ðkÞ

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

¼½A,A

,:::,A

,

(3)

where T

ða

ðkÞÞ ¼ A

¼ I

 a

ðkÞ, i ¼ 1,2, ,n, and I

is q

– 1 -dimensional row vector and its each element is

1, we call A

is q

– 1 clone pool of antibody a

, and q

deﬁned as follows

¼ gðN

,f ða

ðtÞÞ,l

Þ: (4)

In this paper, we set

¼ l



f ða

ðkÞÞ

j¼1

f ða

ðkÞÞ

: (5)

And, we deﬁne q

as follows

¼ Int N



j¼1

, i,j ¼ 1,2, ,n: (6)

> n is a given value related to the scale of clone, and

Int (x) rounds the argument x toward the smallest integer

greater than x. Obviously, q

is used to reﬂect the clonal

scale of antibody a

. And N

i¼1

. In fact, Eq. (6)

shows that each antibody will be viewed locally and has a

different clone size from the others, and its clone size

depends on the antigenic afﬁnity f ða

ðtÞÞ and l

, which is

related to the antibody afﬁnity of a

with other antibodies

in the population (as deﬁned in Eq. (2)).

From the description above, it can be seen that q

directly proportional to l

and the antigenic afﬁnity, which

means the lower the similarity with other antibodies and

the larger the antigenic afﬁnity of antibody a

is, the larger

its clone size is.

Clonal mutation T

: I

! I

. To maintain the

information of the original antibody population, the clonal

mutation does not act on the initial antibodies, A 2 A

which can be understood as follows, for example, one

clonal pool contains 5 of the same antibodies, and then the

clonal mutation will apply to 4 antibodies in this clonal

pool, and leave one antibody as the parent antibody. The

clonal mutation operation can be deﬁned as follows:

ðkÞ¼T

ðA

ðkÞÞ: (7)

Moreover, the mutation probability is used as follows:

pða

! a

Þ¼p

Hða

ð1 – p

L – H ða

, (8)

where H ðx,yÞ is the Hamming distance function.

Clonal selection T

: I

! I

. For the each clone pool

after clonal mutation, select the antibody with the highest

antigenic afﬁnity, namely, b

¼fa

ðkÞjmax f ða

Þ,

j ¼ 1,2, ,q

– 1g, i ¼ 1,2, ,n, if the antibody a

the original population satisﬁes:

f ða

Þ < f ðb

Þ: (9)

Then b

replaces the antibody a

in the original

population. Otherwise, keep the antibody a

. Fig. 1

shows the monoclonal process of one antibody cloned

for 10 times.

5.3 Polyclonal operator

The main difference between the polyclonal operator and

the monoclonal operator lies in that the polyclonal

operator includes clonal recombination.

Clonal recombination T

: I

! I

. Similar to the

clonal mutation, the clonal recombination is unused to the

original population A, and we adopt the following

540 Front. Comput. Sci. China 2010, 4(4): 536–559

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