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RETRACTED ARTICLE: A new sequence optimization algorithm based on particle swarm for machine learning

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This article was retracted on 11 December 2024

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Abstract

With the coming of 5Gera, the algorithm efficiency and accuracy of artificial intelligence and machine learning are facing more strict requirements and greater challenges. Although artificial neural network (ANN) and deep neural network (DNN) do well in classification and regression, a large amount of calculation leads to a slow response. And the speed of sequence maximum optimization (SMO) algorithm is very fast, but not accurate enough. In this paper, research attempts to put forward a sequential optimization algorithm based on the particle swarm optimization to balance the performance of accuracy and response speed. Compared with ANN, DNN and SMO, this approach not only has a high calculation efficiency and but also do well in the classification and regression.

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Acknowledgements

This research was partially supported by Fujian Province Education Hall Youth Project (number: JAT170679), by the Fujian Natural Science Foundation Project (number: 2019J01887), by the Fujian Provincial Marine Economic Development Subsidy Fund Project (number: FJHJF-L-2019-7), by the Electronic information and engineering institute of Fujian Normal University, by the School of Economic of Fujian Normal University, Key Laboratory of Nondestructive Testing, Fuqing Branch of Fujian Normal University;

Funding

This research received no external funding.

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Authors and Affiliations

  1. Electronic Information and Engineering Institute, Fuqing Branch of Fujian Normal University, Fuzhou, China

    Chaofan Xie

  2. Fujian Provincial Key Laboratory of Information Processing and Intelligent Control, Minjiang University, Fuzhou, China

    Fuquan Zhang

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  1. Chaofan Xie
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  2. Fuquan Zhang
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Contributions

Conceptualization, FZ. and CX; methodology, CX; software, CX; validation, FZ and CX; formal analysis, CX; resources, FZ; data curation, CX; writing—original draft preparation, CX; writing—review and editing, CX; visualization, CX; supervision, FZ; project administration, FZ. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Fuquan Zhang.

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The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1007/s12652-024-04943-3

Appendices

Appendix A

The determination of the range of three variables for classification is provided in the following:

Case 1: \(d_{1} = 1,d_{2} = 1,d_{3} = 1\).

  1. 1.

    \(r = 3C\). In the case \(z_{1} = C\)\(z_{{2}} = C\)\(z_{{3}} = C\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

  2. 2.

    \({2}C \le r < 3C\). In the case, the further search possible is shown in the Fig. 14a, feasible solution of convex space is \(\Delta P_{1} P_{2} P_{3}\), the coordinates of P1, P2, P3 are (r − 2C, C, C), (C, r − 2C, C), (C, C, r − 2C).

  3. 3.

    \(C < r < 2C\). In the case, there are six vertexes on boundary. The feasible solution of convex space is shown in the Fig. 14b, P1, P2, P3, P4, P5, P6 are vertex on boundary, the coordinates of P1, P2, P3, P4, P5, P6 are (0, r − C, C), (r − C, 0, C), (C, 0, r − C), (C, r − C, 0), (r − C, C, 0), (0, C, r − C).

  4. 4.

    \({0 < }r \le C\). In the case, the further search possible is shown in the Fig. 14c, the coordinates of P1, P2, P3 are (0, 0, r), (r, 0, 0), (0, r, 0).

  5. 5.

    \(r = {0}\). In the case \(z_{1} = {0}\)\(z_{{2}} = {0}\)\(z_{{3}} = {0}\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

Fig. 14
figure 14

Illustration classification for case 1: a \({2}C \le r < 3C\); b \(C < r < 2C\); c \({0 < }r \le C\)

Full size image

Case 2: \(d_{1} = 1,d_{2} = 1,d_{3} = - 1\).

  1. 1.

    \(r = {2}C\). In the case \(z_{1} = C\)\(z_{{2}} = C\)\(z_{{3}} = {0}\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

  2. 2.

    \(C \le r < {2}C\). In the case, the further search possible is shown in the Fig. 15a, feasible solution of convex space is \(\Delta P_{1} P_{2} P_{3}\), the coordinates of P1, P2, P3 are (C, C, 2C − r), (C, r − C, 0), (r − C, C, 0).

  3. 3.

    \({0} < r < C\). In the case, there are six vertexes on boundary. The feasible solution of convex space is shown in the Fig. 15b, P1, P2, P3, P4, P5, P6 are vertex on boundary, the coordinates of P1, P2, P3, P4, P5, P6 are (r, C, C), (C, r, C), (C, 0, C − r), (r, 0, 0), (0, r, 0), (0, C, C − r).

  4. 4.

    \(- C{ < }r \le 0\). In the case, the further search possible is shown in the Fig. 15c, the coordinates of P1, P2, P3 are (0, r + C, C), (r + C, 0, C), (0, 0, − r).

  5. 5.

    \(r = - C\). In the case \(z_{1} = {0}\)\(z_{{2}} = {0}\)\(z_{{3}} = C\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

Fig. 15
figure 15

Illustration classification for case 2: a \(C \le r < {2}C\); b \({0} < r < C\); c \(- C{ < }r \le 0\)

Full size image

Case 3: \(d_{1} = 1,d_{2} = - 1,d_{3} = 1\)

  1. 1.

    \(r = {2}C\). In the case \(z_{1} = C\)\(z_{{2}} = {0}\)\(z_{{3}} = C\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

  2. 2.

    \(C \le r < {2}C\). In the case, the further search possible is shown in the Fig. 16a, feasible solution of convex space is \(\Delta P_{1} P_{2} P_{3}\), the coordinates of P1, P2, P3 are (r − C, 0, C), (C, 0, r − C), (C, 2C − r, C).

  3. 3.

    \({0} < r < C\). In the case, there are six vertexes on boundary. The feasible solution of convex space is shown in the Fig. 16b P1, P2, P3, P4, P5, P6 are vertex on boundary, the coordinates of P1, P2, P3, P4, P5, P6 are (0, C − r, C), (0, 0, r), (r, 0, 0), (C, C − r, 0), (C, C, r), (r, C, C).

  4. 4.

    \(- C{ < }r \le 0\). In the case, the further search possible is shown in the Fig. 16c, the coordinates of P1, P2, P3 are (0, C, r + C), (0, − r, 0), (r + C, C, 0).

  5. 5.

    \(r = - C\). In the case \(z_{1} = {0}\)\(z_{{2}} = C\)\(z_{{3}} = 0\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

Fig. 16
figure 16

Illustration classification for case 3: a \(C \le r < {2}C\); b \({0} < r < C\); c \(- C{ < }r \le 0\)

Full size image

Case 4: \(d_{1} = - 1,d_{2} = 1,d_{3} = 1\).

  1. 1.

    \(r = {2}C\). In the case \(z_{1} = 0\)\(z_{{2}} = C\)\(z_{{3}} = C\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

  2. 2.

    \(C \le r < {2}C\). In the case, the further search possible is shown in the Fig. 17a, feasible solution of convex space is \(\Delta P_{1} P_{2} P_{3}\), the coordinates of P1, P2, P3 are (0, r − C, C), (2C−r, C, C), (0, C, r − C).

  3. 3.

    \({0} < r < C\). In the case, there are six vertexes on boundary. The feasible solution of convex space is shown in the Fig. 17b, P1, P2, P3, P4, P5, P6 are vertex on boundary, the coordinates of P1, P2, P3, P4, P5, P6 are (C − r, 0, C), (C, r, C), (C, C, r), (C − r, C, 0) (0, r, 0), (0, 0, r).

  4. 4.

    \(- C{ < }r \le 0\). In the case, the further search possible is shown in the Fig. 17c, the coordinates of P1, P2, P3 are (C, 0, r + C), (− r, 0, 0), (C, r + C, 0).

  5. 5.

    \(r = - C\). In the case \(z_{1} = C\)\(z_{{2}} = 0\)\(z_{{3}} = 0\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

Fig. 17
figure 17

Illustration classification for case 4: a \(C \le r < {2}C\); b \({0} < r < C\); c \(- C{ < }r \le 0\)

Full size image

Case 5: \(d_{1} = 1,d_{2} = - 1,d_{3} = - 1\).

Due to \(r = z_{1} - z_{2} - z_{3}\), multiply − 1 on both sides can get \(- r = - z_{1} + z_{2} + z_{3}\), and \(- C \le - r \le 2C\), so it transform into Case 4.

Case 6: \(d_{1} = - 1,d_{2} = 1,d_{3} = - 1\).

Due to \(r = - z_{1} + z_{2} - z_{3}\), multiply − 1 on both sides can get \(- r = z_{1} - z_{2} + z_{3}\), and \(- C \le - r \le 2C\), so it transform into Case 3.

Case 7: \(d_{1} = - 1,d_{2} = - 1,d_{3} = 1\).

Due to \(r = - z_{1} - z_{2} + z_{3}\), multiply − 1 on both sides can get \(- r = z_{1} + z_{2} - z_{3}\), and \(- C \le - r \le 2C\), so it transform into Case 2.

Case 8: \(d_{1} = - 1,d_{2} = - 1,d_{3} = - 1\).

Due to \(r = - z_{1} - z_{2} - z_{3}\), multiply − 1 on both sides can get \(- r = z_{1} + z_{2} + z_{3}\), and \({0} \le - r \le {3}C\), so it transform into Case 1.

For example, if you select four variables as optimization.

Case 1: \(d_{1} = 1,d_{2} = 1,d_{3} = 1,d_{{4}} = 1\).

  1. 1.

    \(r = {4}C\). In the case \(z_{1} = C\)\(z_{{2}} = C\)\(z_{{3}} = C\)\(z_{{4}} = C\).

  2. 2.

    \({3}C \le r < {4}C\). The set of convex vertices are (r − 3C, C, C, C), (C, r − 3C, C, C), (C, C, r − 3C, C), (C, C, C, r − 3C).

  3. 3.

    \({2}C < r < {3}C\). The set of convex vertices are (0, r − 2C, C, C), (r − 2C, 0, C, C), (0, C, r − 2C, C), (r − 2C, C, 0, C), (0, C, C, r − 2C), (r − 2C, C, C, 0), (C, 0, r − 2C, C), (C, r − 2C, 0, C), (C, 0, C, r − 2C), (C, r − 2C, C, 0), (C, C, 0, r − 2C), (C, C, r − 2C, 0).

  4. 4.

    \({\text{C < }}r < 2C\). The set of convex vertices are (0, 0, r − C, C), (0, 0, C, r − C), (0, C, 0, r − C), (0, r − C, 0, C), (0, C, r − C, 0), (0, r − C, C, 0), (C, 0, 0, r − C), (r − C, 0, 0, C), (C, 0, r − C, 0), (r − C, 0, C, 0), (r − C, C, 0, 0), (C, r − C, 0, 0).

  5. 5.

    \({0 < }r \le C\). The set of convex vertices are (0, 0, 0, r), (r, 0, 0, 0), (0, r, 0, 0), (0, 0, r, 0).

  6. 6.

    \(r = {0}\). In the case \(z_{1} = {0}\)\(z_{{2}} = {0}\)\(z_{{3}} = {0}\)\(z_{{4}} = {0}\)

Case 2: \(d_{1} = 1,d_{2} = 1,d_{3} = 1,d_{{4}} = { - }1\).

  1. 1.

    \(r = {3}C\). In the case \(z_{1} = C\)\(z_{{2}} = C\)\(z_{{3}} = C\)\(z_{{4}} = {0}\)

  2. 2.

    \({2}C \le r < {3}C\). The set of convex vertices are (C, C, C, 3C − r), (C, C, r − 2C, 0), (C, r − 2C, C, 0), (r − 2C, C, C, 0).

  3. 3.

    \(C \le r < 2C\). The set of convex vertices are(0, r − C, C, 0), (r − C, 0, C, 0), (C, 0, r –C, 0), (C, r − C, 0, 0), (r − C, C, 0, 0), (0, C, r − C, 0), (0, r − C, C, C), (r − C, 0, C, C), (C, C, r − C, C), (0, C, C, 2C − r), (C, C, 0, 2C − r), (C, 0, C, 2C − r)

  4. 4.

    \({0} \le r < C\). The set of convex vertices are (0, 0, r, 0), (r, 0, 0, 0), (0, r, 0, 0), (0, r, C, C), (r, 0, C, C), (C, 0, r, C), (C, r, 0, C), (r, C, 0, C), (0, C, r, C), (C, 0, 0, C − r), (0, C, 0, C − r), (0, 0, C, C − r)

  5. 5.

    \(- C < r < 0\). The set of convex vertices are (r + C, 0, 0, C), (0, r + C, 0, C), (0, 0, r + C, C), (0, 0, 0, − r)

  6. 6.

    \(r = - C\). In the case \(z_{1} = {0}\)\(z_{{2}} = {0}\)\(z_{{3}} = {0}\)\(z_{{4}} = C\).

Case 3: \(d_{1} = 1,d_{2} = 1,d_{3} = - 1,d_{{4}} = - 1\).

  1. 1.

    \(r = {2}C\). In the case \(z_{1} = C\)\(z_{{2}} = C\)\(z_{{3}} = {0}\)\(z_{{4}} = {0}\)

  2. 2.

    \(C \le r < {2}C\). The set of convex vertices are (C, r, C, 0), (r, C, C, 0), (C, r, 0, C), (r, C, 0, C).

  3. 3.

    \(0 \le r < C\). The set of convex vertices are(C, C, C, C − r), (0, C, 0, C − r), (C, 0, 0, C − r), (C, C, C − r, C), (0, C, C − r, 0), (C, 0, C − r, 0), (r, 0, 0, 0), (0, r, 0, 0), (r, C, C, 0), (r, C, 0, C), (C, r, C, 0), (C, r, 0, C)

  4. 4.

    \(- C \le r < 0\). The set of convex vertices are (C + r, C, C, C), (C + r, 0, 0, C), (C + r, 0, C, 0), (C, C + r, C, C), (0, C + r, 0, C), (0, C + r, C, 0), (0, 0, − r, 0), (C, 0, − r, C), (0, C, − r, C), (0, 0, 0, − r), (C, 0, C, − r), (0, C, C, − r)

  5. 5.

    \(- 2C < r < - C\). The set of convex vertices are (2C + r, 0, C, C), (0, 2C + r, C, C), (0, 0, − r − C, C), (0, 0, C, − r − C)

  6. 6.

    \(r = - {2}C\). In the case \(z_{1} = {0}\)\(z_{{2}} = {0}\)\(z_{{3}} = C\)\(z_{{4}} = C\).

Other case can transform into above case which is similar with three parameters.

Appendix B

The determination of the range of three variables for regression is provided in the following:

Case 1: \(r = 3C\).In the case \(z_{1} = C\)\(z_{{2}} = C\)\(z_{{3}} = C\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

Case 2: \(C \le r < 3C\).In the case, the further search possible is shown in the Fig. 18, feasible solution of convex space is \(\Delta P_{1} P_{2} P_{3}\), P1, P2, P3 are vertex on boundary, the coordinates of P1, P2, P3 are (r − 2C, C, C), (C, r − 2C, C), (C, C, r − 2C).

Case 3: \(- C \le r < C\).In the case, there are six vertexes on boundary. The feasible solution of convex space is shown in the Fig. 19P1, P2, P3, P4, P5, P6 are vertex on boundary, the coordinates of P1, P2, P3, P4, P5, P6 are (− C, r, C), (− C, C, r), (r, C, − C), (C, r, − C), (C, − C, r), (r, − C, C).

Case 4: \(- {3}C < r \le - C\).In the case, the feasible solution of convex space is shown in the Fig. 20, P1, P2, P3 are vertex on boundary, the coordinates of P1, P2, P3 are (− C, − C, r), (− C, r, − C), (r, − C, − C).

Fig. 18
figure 18

Illustration regression for case 2 \(C \le r < 3C\)

Full size image
Fig. 19
figure 19

Illustration regression for case 3 \(- C \le r < C\)

Full size image
Fig. 20
figure 20

Illustration regression for case 4 \(- {3}C < r \le - C\)

Full size image

Case 5: \(r = - 3C\). In the case, \(z_{1} = - C\) \(z_{{2}} = - C\) \(z_{{3}} = - C\). No further search is possible and we may find another to continue the algorithm or simply ignore this case.

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Xie, C., Zhang, F. RETRACTED ARTICLE: A new sequence optimization algorithm based on particle swarm for machine learning. J Ambient Intell Human Comput 13, 2601–2619 (2022). https://doi.org/10.1007/s12652-021-03004-3

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