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Simulated annealing-based immunodominance algorithm for multi-objective optimization problems

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Abstract

Based on the simulated annealing strategy and immunodominance in the artificial immune system, a simulated annealing-based immunodominance algorithm (SAIA) for multi-objective optimization (MOO) is proposed in this paper. In SAIA, all immunodominant antibodies are divided into two classes: the active antibodies and the hibernate antibodies at each temperature. Clonal proliferation and recombination are employed to enhance local search on those active antibodies while the hibernate antibodies have no function, but they could become active during the following temperature. Thus, all antibodies in the search space can be exploited effectively and sufficiently. Simulated annealing-based adaptive hypermutation, population pruning, and simulated annealing selection are proposed in SAIA to evolve and obtain a set of antibodies as the trade-off solutions. Complexity analysis of SAIA is also provided. The performance comparison of SAIA with some state-of-the-art MOO algorithms in solving 14 well-known multi-objective optimization problems (MOPs) including four many objectives test problems and twelve multi-objective 0/1 knapsack problems shows that SAIA is superior in converging to approximate Pareto front with a standout distribution.

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Acknowledgements

The authors would like to thank the editor and the reviewers for helpful comments that greatly improved the paper. We gratefully acknowledge our colleague, Prof. J Liu, who helps us to modify and polish English writing. This work was supported by the National Natural Science Foundation of China (No. 61373111); the Fundamental Research Funds for the Central University (Nos. K50511020014, K5051302084, JB150227); the Provincial Natural Science Foundation of Shaanxi of China (No. 2014JM8321).

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Authors and Affiliations

  1. Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an, 710071, China

    Ruochen Liu, Jianxia Li, Xiaolin Song, Xin Yu & Licheng Jiao

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Appendix

Appendix

SCH:

$$\begin{aligned} f_1 (x)=\left\{ {{\begin{array}{l} {-x, if x \le 1} \\ {-2+x, if 1< x<3} \\ {4-x, if 3< x < 4} \\ {-4+x, if x > 4} \\ \end{array} },f_2 (x)=(x-5)^{2}} \right. \end{aligned}$$

where \(x\in [-5,10]\).

DEB:

$$\begin{aligned} f_1 (x)=x_1 ,f_2 (x)=(1+10x_2 )\times \left[ 1-\left( \frac{x_1 }{1+10x_2 }\right) ^{2}-\frac{x_1 }{1+10x_2 }\sin (8\pi x_1 )\right] \end{aligned}$$

where \(x=(x_{1},x_{2})\in [0,1]\).

ZDT1:

$$\begin{aligned} f_1 (x)=x_1 ,f_2 (x)=g(x)\left[ {1-\sqrt{\frac{f_1 (x)}{g(x)}}} \right] \end{aligned}$$

where \(g(x)=1+{9\sum _{i=2}^l {x_i } }/{\left( {l-1} \right) }\), and \(x=(x_{1},{\ldots },x_{l})^{T}\in [0,1]^{l}\).

ZDT2:

$$\begin{aligned} f_1 (x)=x_1 ,\;f_2 (x)=g(x)\left[ {1-\left( {\frac{f_1 (x)}{g(x)}} \right) ^{2}} \right] \end{aligned}$$

where g(x) and the range of x are the same as that of ZDT1.

ZDT3:

$$\begin{aligned} f_1 (x)=x_1 ,\quad f_2 (x)=g(x)\left[ {1-\sqrt{\frac{f_1 (x)}{g(x)}}} \right] -\frac{f_1 (x)}{g(x)}\sin (10\pi x_1 ) \end{aligned}$$

where g(x) and the range of x are also the same as that of ZDT1.

DTLZ1:

$$\begin{aligned} \begin{array}{c} f_1 (x)=0.5x_1 x_2 \ldots x_{k-1} \left( {1+g(x_k )} \right) \\ f_2 (x)=0.5x_1 x_2 \ldots \left( {1-x_{k-1} } \right) \left( {1+g(x_k )} \right) \\ \ldots \\ f_{k-1} (x)=0.5x_1 \left( {1-x_2 } \right) \left( {1+g(x_k )} \right) \\ f_k (x)=0.5\left( {1-x_1 } \right) \left( {1+g(x_k )} \right) \\ \end{array} \end{aligned}$$

where \(g(x)=100\left[ {\left| {x_k } \right| +\sum _{x_i \in x_k } {\left( {\left( {x-0.5} \right) ^{2}-\cos \left( {20\pi \left( {x-0.5} \right) } \right) } \right) } } \right] \), and \(x=(x_{1},{\ldots },x_{l})^{T}\in [0,1]^{l}\).

DTLZ2:

$$\begin{aligned} \begin{array}{c} f_1 (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 \pi } \right) \ldots \cos \left( {0.5x_{k-1} \pi } \right) \\ f_2 (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 \pi } \right) \ldots \sin \left( {0.5x_{k-1} \pi } \right) \\ \ldots \\ f_{k-1} (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 \pi } \right) \sin \left( {0.5x_2 \pi } \right) \\ f_k (x)=\left( {1+g(x_k )} \right) \sin \left( {0.5x_1 \pi } \right) \\ \end{array} \end{aligned}$$

where \(g(x)=\sum _{x_i \in x_k } {\left( {x-0.5} \right) ^{2}} \), and \(x=(x_{1},{\ldots },x_{l})^{T}\in [0,1]^{l}\).

DTLZ3:

$$\begin{aligned} \begin{array}{c} f_1 (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 \pi } \right) \ldots \cos \left( {0.5x_{k-1} \pi } \right) \\ f_2 (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 \pi } \right) \ldots \sin \left( {0.5x_{k-1} \pi } \right) \\ \ldots \\ f_{k-1} (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 \pi } \right) \sin \left( {0.5x_2 \pi } \right) \\ f_k (x)=\left( {1+g(x_k )} \right) \sin \left( {0.5x_1 \pi } \right) \\ \end{array} \end{aligned}$$

where \(g(x)=100\left[ {\left| {x_k } \right| +\sum _{x_i \in x_k } {\left( {\left( {x-0.5} \right) ^{2}-\cos \left( {20\pi \left( {x-0.5} \right) } \right) } \right) } } \right] \), and \(x= (x_{1},{\ldots },x_{l})^{T}\in [0,1]^{l}\).

DTLZ4:

$$\begin{aligned} \begin{array}{c} f_1 (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 ^{\alpha }\pi } \right) \ldots \cos \left( {0.5x_{k-1} ^{\alpha }\pi } \right) \\ f_2 (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 ^{\alpha }\pi } \right) \ldots \sin \left( {0.5x_{k-1} ^{\alpha }\pi } \right) \\ \ldots \\ f_{k-1} (x)=\left( {1+g(x_k )} \right) \cos \left( {0.5x_1 ^{\alpha }\pi } \right) \sin \left( {0.5x_2 ^{\alpha }\pi } \right) \\ f_k (x)=\left( {1+g(x_k )} \right) \sin \left( {0.5x_1 ^{\alpha }\pi } \right) \\ \end{array} \end{aligned}$$

where \(g(x)=\sum _{x_i \in x_k } {\left( {x-0.5} \right) ^{2}} ,\alpha =100\), and \(x=(x_{1},{\ldots },x_{l})^{T}\in [0,1]^{l}\).

DTLZ6:

$$\begin{aligned} \begin{array}{c} f_1 (x)=x_1 \\ f_2 (x)=x_2 \\ \cdots \\ f_{k-1} (x)=x_{k-1} \\ f_k (x)=\left( {1+g(x_k )} \right) h\left( {f_1 ,f_2 ,\ldots ,f_{k-1} ,g} \right) \\ \end{array} \end{aligned}$$

where \(h=k-\sum _{i=1}^k {\left[ {\frac{f_i }{1+g}\left( {1+\sin \left( {3\pi f_i } \right) } \right) } \right] } ,g(x)=1+\frac{9}{\left| {x_k } \right| }\sum _{x_i \in x_k } {x_i } \), and \(x=(x_{1},{\ldots },x_{l})^{T}\in [0,1]^{l}\).

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Liu, R., Li, J., Song, X. et al. Simulated annealing-based immunodominance algorithm for multi-objective optimization problems. Knowl Inf Syst 55, 215–251 (2018). https://doi.org/10.1007/s10115-017-1065-x

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