Research Article

Adaptive Particle Search with Operator-Entropy Feedback: A Conditional Framework and Reported Benchmark Summaries

Authors

Seyyed Ahmad Edalatpanah * Department of Applied Mathematics,Ayandegan University, Tonekabon, Iran.
Darko Bozanic University of Defence in Belgrade, Military Academy.
Natalja Osintsev Fraunhofer-Institut für Holzforschung Wilhelm-Klauditz Institut WKI, Bienroder Weg 54 E, 38108 Brunswick, Germany.

https://doi.org/10.22105/opt.v3i4.124

Downloads

Abstract

Population search can combine particle motion, adaptive operator selection and constraint penalties, but its guarantees depend on the proposal kernels and evaluation protocol. This paper analyzes such a framework, identified as MAE-HBA in the retained illustrations, for bounded continuous optimization. The specified motion is a particle-swarm update; operator-selection entropy measures the distribution of operator choices rather than the spatial diversity of particles. We distinguish this specification from an implemented operator portfolio and derive a conditional coverage bound for a fixed, exactly evaluated objective. Local stability is examined for a frozen scalar particle update, without extending that result to the stochastic adaptive population. Reported benchmark and ablation summaries are retained, but their experiment identity is unresolved: the same numerical tables occur in a companion manuscript, and run-level outputs and implementation records are unavailable for this analysis. The summaries therefore provide descriptive comparisons rather than independent empirical validation. The framework does not yet establish multimodal adapter fine-tuning, manifold optimization, calibrated adversarial robustness or global convergence of an executed algorithm.

Keywords

Particle swarm optimization Adaptive operator selection Operator entropy Constraint handling Conditional convergence Benchmark provenance

References (76)

[1] Eberhart, R., & Kennedy, J. (1995). Particle swarm optimization. Proceedings of the IEEE international conference on neural networks (Vol. 4, pp. 1942–1948). IEEE. https://doi.org/10.1109/ICNN.1995.488968

[2] Holland, J. H. (1992). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. MIT press. https://doi.org/10.7551/mitpress/1090.001.0001

[3] Storn, R., & Price, K. (1997). Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341–359. https://doi.org/10.1023/A:1008202821328

[4] Črepinšek, M., Liu, S. H., & Mernik, M. (2013). Exploration and exploitation in evolutionary algorithms: A survey. ACM computing surveys (CSUR), 45(3), 1–33. https://doi.org/10.1145/2480741.2480752

[5] Hellwig, M., & Beyer, H. G. (2019). Benchmarking evolutionary algorithms for single objective real-valued constrained optimization – A critical review. Swarm and evolutionary computation, 44, 927–944. https://doi.org/10.1016/j.swevo.2018.10.002

[6] Kar, A. K. (2016). Bio inspired computing – A review of algorithms and scope of applications. Expert systems with applications, 59, 20–32. https://doi.org/10.1016/j.eswa.2016.04.018

[7] Gendreau, M., & Potvin, J. Y. (2005). Metaheuristics in combinatorial optimization. Annals of operations research, 140(1), 189–213. https://doi.org/10.1007/s10479-005-3971-7

[8] Talbi, E. G. (2009). From design to implementation. Wiley. https://doi.org/10.1002/9780470496916

[9] Boussaïd, I., Lepagnot, J., & Siarry, P. (2013). A survey on optimization metaheuristics. Information sciences, 237, 82–117. https://doi.org/10.1016/j.ins.2013.02.041

[10] Advanced Evolutionary Algorithms. (2010). In Introduction to evolutionary algorithms (pp. 39–132). London: Springer London. https://doi.org/10.1007/978-1-84996-129-5_3

[11] Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007

[12] Mirjalili, S., & Lewis, A. (2016). The whale optimization algorithm. Advances in engineering software, 95, 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008

[13] Mirjalili, S., Gandomi, A. H., Mirjalili, S. Z., Saremi, S., Faris, H., & Mirjalili, S. M. (2017). Salp swarm algorithm: A bio-inspired optimizer for engineering design problems. Advances in engineering software, 114, 163–191. https://doi.org/10.1016/j.advengsoft.2017.07.002

[14] Heidari, A. A., Mirjalili, S., Faris, H., Aljarah, I., Mafarja, M., & Chen, H. (2019). Harris hawks optimization: Algorithm and applications. Future generation computer systems, 97, 849–872. https://doi.org/10.1016/j.future.2019.02.028

[15] Tanabe, R., & Fukunaga, A. (2013). Success-history based parameter adaptation for differential evolution. 2013 IEEE congress on evolutionary computation (pp. 71–78). IEEE. https://doi.org/10.1109/CEC.2013.6557555

[16] Hansen, N. (2006). The CMA Evolution Strategy: A Comparing Review. In Towards a new evolutionary computation: Advances in the estimation of distribution algorithms (pp. 75–102). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-32494-1_4

[17] Wolpert, D. H., & Macready, W. G. (2002). No free lunch theorems for optimization. IEEE transactions on evolutionary computation, 1(1), 67–82. https://doi.org/10.1109/4235.585893

[18] Wu, G., Mallipeddi, R., & Suganthan, P. N. (2017). Problem definitions and evaluation criteria for the CEC 2017 competition on constrained real-parameter optimization. https://github.com/P-N-Suganthan/CEC2017-BoundContrained

[19] Kumar, A., Price, K. V, Mohamed, A. W., Hadi, A. A., & Suganthan, P. N. (2021). Problem definitions and evaluation criteria for the CEC 2022 special session and competition on single objective bound constrained numerical optimization. https://github.com/P-N-Suganthan/2022-SO-BO

[20] Demšar, J. (2006). Statistical comparisons of classifiers over multiple data sets. The journal of machine learning research, 7, 1–30. https://doi.org/10.5555/1248547.1248548

[21] Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American statistical association, 32(200), 675–701. https://doi.org/10.1080/01621459.1937.10503522

[22] Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics bulletin, 1(6), 80–83. https://doi.org/10.2307/3001968

[23] Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, 65–70. https://doi.org/10.2307/4615733

[24] Derrac, J., García, S., Molina, D., & Herrera, F. (2011). A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm and evolutionary computation, 1(1), 3–18. https://doi.org/10.1016/j.swevo.2011.02.002

[25] Saltelli, A. (2002). Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2), 280–297. https://doi.org/10.1016/S0010-4655(02)00280-1

[26] Sobol′, I. M. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and computers in simulation, 55(1), 271–280. https://doi.org/10.1016/S0378-4754(00)00270-6

[27] Morris, M. D. (1991). Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2), 161–174. https://doi.org/10.1080/00401706.1991.10484804

[28] Bergstra, J., Bardenet, R., Bengio, Y., & Kégl, B. (2011). Algorithms for hyper-parameter optimization. Advances in neural information processing systems (pp. 1–9). Curran Associates, Inc. https://doi.org/10.5555/2986459.2986743

[29] Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical bayesian optimization of machine learning algorithms. Advances in neural information processing systems (Vol. 25, pp. 1–9). Curran Associates, Inc. https://doi.org/10.5555/2999325.2999464

[30] Akiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. (2019). Optuna: A next-generation hyperparameter optimization framework. Proceedings of the 25th ACM Sigkdd international conference on knowledge discovery & data mining (pp. 2623–2631). New York, NY, USA: Association for Computing Machinery. 10.1145/3292500.3330701

[31] Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2), 182–197. https://doi.org/10.1109/4235.996017

[32] Got, A., Moussaoui, A., & Zouache, D. (2020). A guided population archive whale optimization algorithm for solving multiobjective optimization problems. Expert systems with applications, 141, 112972. https://doi.org/10.1016/j.eswa.2019.112972

[33] Brammya, G., Praveena, S., Ninu Preetha, N. S., Ramya, R., Rajakumar, B. R., & Binu, D. (2019). Deer hunting optimization algorithm: A new nature-inspired meta-heuristic paradigm. The computer journal, bxy133. https://doi.org/10.1093/comjnl/bxy133

[34] Daliri, A., Asghari, A., Azgomi, H., & Alimoradi, M. (2022). The water optimization algorithm: A novel metaheuristic for solving optimization problems. Applied intelligence, 52(15), 17990–18029. https://doi.org/10.1007/s10489-022-03397-4

[35] Alimoradi, M., Azgomi, H., & Asghari, A. (2022). Trees social relations optimization algorithm: A new swarm-based metaheuristic technique to solve continuous and discrete optimization problems. Mathematics and computers in simulation, 194, 629–664. https://doi.org/10.1016/j.matcom.2021.12.010

[36] Lundberg, S. M., & Lee, S. I. (2017). A unified approach to interpreting model predictions. NIPS'17: Proceedings of the 31st international conference on neural information processing systems. (Vol. 30, pp. 4768–4777). NIPS. https://doi.org/10.5555/3295222.3295230

[37] Ribeir

[38] Eberhart, R., & Kennedy, J. (1995). Particle swarm optimization. Proceedings of the IEEE international conference on neural networks (Vol. 4, pp. 1942–1948). IEEE. https://doi.org/10.1109/ICNN.1995.488968

[39] Holland, J. H. (1992). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. MIT press. https://doi.org/10.7551/mitpress/1090.001.0001

[40] Storn, R., & Price, K. (1997). Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341–359. https://doi.org/10.1023/A:1008202821328

[41] Črepinšek, M., Liu, S. H., & Mernik, M. (2013). Exploration and exploitation in evolutionary algorithms: A survey. ACM computing surveys (CSUR), 45(3), 1–33. https://doi.org/10.1145/2480741.2480752

[42] Hellwig, M., & Beyer, H. G. (2019). Benchmarking evolutionary algorithms for single objective real-valued constrained optimization – A critical review. Swarm and evolutionary computation, 44, 927–944. https://doi.org/10.1016/j.swevo.2018.10.002

[43] Kar, A. K. (2016). Bio inspired computing – A review of algorithms and scope of applications. Expert systems with applications, 59, 20–32. https://doi.org/10.1016/j.eswa.2016.04.018

[44] Gendreau, M., & Potvin, J. Y. (2005). Metaheuristics in combinatorial optimization. Annals of operations research, 140(1), 189–213. https://doi.org/10.1007/s10479-005-3971-7

[45] Talbi, E. G. (2009). From design to implementation. Wiley. https://doi.org/10.1002/9780470496916

[46] Boussaïd, I., Lepagnot, J., & Siarry, P. (2013). A survey on optimization metaheuristics. Information sciences, 237, 82–117. https://doi.org/10.1016/j.ins.2013.02.041

[47] Advanced Evolutionary Algorithms. (2010). In Introduction to evolutionary algorithms (pp. 39–132). London: Springer London. https://doi.org/10.1007/978-1-84996-129-5_3

[48] Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007

[49] Mirjalili, S., & Lewis, A. (2016). The whale optimization algorithm. Advances in engineering software, 95, 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008

[50] Mirjalili, S., Gandomi, A. H., Mirjalili, S. Z., Saremi, S., Faris, H., & Mirjalili, S. M. (2017). Salp swarm algorithm: A bio-inspired optimizer for engineering design problems. Advances in engineering software, 114, 163–191. https://doi.org/10.1016/j.advengsoft.2017.07.002

[51] Heidari, A. A., Mirjalili, S., Faris, H., Aljarah, I., Mafarja, M., & Chen, H. (2019). Harris hawks optimization: Algorithm and applications. Future generation computer systems, 97, 849–872. https://doi.org/10.1016/j.future.2019.02.028

[52] Tanabe, R., & Fukunaga, A. (2013). Success-history based parameter adaptation for differential evolution. 2013 IEEE congress on evolutionary computation (pp. 71–78). IEEE. https://doi.org/10.1109/CEC.2013.6557555

[53] Hansen, N. (2006). The CMA Evolution Strategy: A Comparing Review. In Towards a new evolutionary computation: Advances in the estimation of distribution algorithms (pp. 75–102). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-32494-1_4

[54] Wolpert, D. H., & Macready, W. G. (2002). No free lunch theorems for optimization. IEEE transactions on evolutionary computation, 1(1), 67–82. https://doi.org/10.1109/4235.585893

[55] Wu, G., Mallipeddi, R., & Suganthan, P. N. (2017). Problem definitions and evaluation criteria for the CEC 2017 competition on constrained real-parameter optimization. https://github.com/P-N-Suganthan/CEC2017-BoundContrained

[56] Kumar, A., Price, K. V, Mohamed, A. W., Hadi, A. A., & Suganthan, P. N. (2021). Problem definitions and evaluation criteria for the CEC 2022 special session and competition on single objective bound constrained numerical optimization. https://github.com/P-N-Suganthan/2022-SO-BO

[57] Demšar, J. (2006). Statistical comparisons of classifiers over multiple data sets. The journal of machine learning research, 7, 1–30. https://doi.org/10.5555/1248547.1248548

[58] Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American statistical association, 32(200), 675–701. https://doi.org/10.1080/01621459.1937.10503522

[59] Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics bulletin, 1(6), 80–83. https://doi.org/10.2307/3001968

[60] Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, 65–70. https://doi.org/10.2307/4615733

[61] Derrac, J., García, S., Molina, D., & Herrera, F. (2011). A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm and evolutionary computation, 1(1), 3–18. https://doi.org/10.1016/j.swevo.2011.02.002

[62] Saltelli, A. (2002). Making best use of model evaluations to compute sensitivity indices. Computer physics communications, 145(2), 280–297. https://doi.org/10.1016/S0010-4655(02)00280-1

[63] Sobol′, I. M. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and computers in simulation, 55(1), 271–280. https://doi.org/10.1016/S0378-4754(00)00270-6

[64] Morris, M. D. (1991). Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2), 161–174. https://doi.org/10.1080/00401706.1991.10484804

[65] Bergstra, J., Bardenet, R., Bengio, Y., & Kégl, B. (2011). Algorithms for hyper-parameter optimization. Advances in neural information processing systems (pp. 1–9). Curran Associates, Inc. https://doi.org/10.5555/2986459.2986743

[66] Snoek, J., Larochelle, H., & Adams, R. P. (2012). Practical bayesian optimization of machine learning algorithms. Advances in neural information processing systems (Vol. 25, pp. 1–9). Curran Associates, Inc. https://doi.org/10.5555/2999325.2999464

[67] Akiba, T., Sano, S., Yanase, T., Ohta, T., & Koyama, M. (2019). Optuna: A next-generation hyperparameter optimization framework. Proceedings of the 25th ACM Sigkdd international conference on knowledge discovery & data mining (pp. 2623–2631). New York, NY, USA: Association for Computing Machinery. 10.1145/3292500.3330701

[68] Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2), 182–197. https://doi.org/10.1109/4235.996017

[69] Got, A., Moussaoui, A., & Zouache, D. (2020). A guided population archive whale optimization algorithm for solving multiobjective optimization problems. Expert systems with applications, 141, 112972. https://doi.org/10.1016/j.eswa.2019.112972

[70] Brammya, G., Praveena, S., Ninu Preetha, N. S., Ramya, R., Rajakumar, B. R., & Binu, D. (2019). Deer hunting optimization algorithm: A new nature-inspired meta-heuristic paradigm. The computer journal, bxy133. https://doi.org/10.1093/comjnl/bxy133

[71] Daliri, A., Asghari, A., Azgomi, H., & Alimoradi, M. (2022). The water optimization algorithm: A novel metaheuristic for solving optimization problems. Applied intelligence, 52(15), 17990–18029. https://doi.org/10.1007/s10489-022-03397-4

[72] Alimoradi, M., Azgomi, H., & Asghari, A. (2022). Trees social relations optimization algorithm: A new swarm-based metaheuristic technique to solve continuous and discrete optimization problems. Mathematics and computers in simulation, 194, 629–664. https://doi.org/10.1016/j.matcom.2021.12.010

[73] Lundberg, S. M., & Lee, S. I. (2017). A unified approach to interpreting model predictions. 31st conference on neural information processing systems (NIPS 2017), Long Beach, CA, USA. (Vol. 30, pp. 4768–4777). NIPS. https://doi.org/10.5555/3295222.3295230

[74] Ribeiro, M. T., Singh, S., & Guestrin, C. (2016). “Why should i trust you?”: Explaining the predictions of any classifier. Proceedings of the 22nd ACM Sigkdd international conference on knowledge discovery and data mining (pp. 1135–1144). New York, NY, USA: Association for Computing Machinery. https://doi.org/10.1145/2939672.2939778

[75] o, M. T., Singh, S., & Guestrin, C. (2016). “Why should i trust you?”: Explaining the predictions of any classifier. Proceedings of the 22nd ACM Sigkdd international conference on knowledge discovery and data mining (pp. 1135–1144). New York, NY, USA: Association for Computing Machinery. https://doi.org/10.1145/2939672.2939778

[76]

Article Metrics

Published 2026-12-15
Issue Vol. 3 No. 4 (2026): Optimality
Section Articles
Document Type Research Article
Views 0
Downloads 0

License

Copyright (c) 2026 Optimality

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

View License Details

How to Cite

Edalatpanah, S. A. ., Bozanic, D. ., & Osintsev, N. . (2026). Adaptive Particle Search with Operator-Entropy Feedback: A Conditional Framework and Reported Benchmark Summaries. Optimality, 3(4), 341-356. https://doi.org/10.22105/opt.v3i4.124

Download Citation

Similar Articles

Mei-Lin Nakamura, Pavel D. Sokolov, Adaeze O. Okonkwo, Adaptive Operator Portfolio Search with Fold Based Landscape Probing , Optimality: Vol. 3 No. 3 (2026): Optimality
Fatemeh Ebrahimzadeh, Peyman Rivandi, Transparency in Population-Based Search: A Methodological Review of Interpretable Search Components , Optimality: Vol. 3 No. 3 (2026): Optimality
Khan Hassan M. , Ahmed M. El-Sayed, Rania Boutros, Tomasz Lewandowski, Constraint-Aware Entropy-Adaptive Swarm Search for Civil Infrastructure Sizing: Formulation and Reported Benchmark Evidence , Optimality: Vol. 3 No. 3 (2026): Optimality
Bilal Ahmad Khan, Wolfgang H. Müller, Distributed Influence Search for Engineering Optimization: Conditional Analysis and Reported Benchmarks , Optimality: Vol. 3 No. 3 (2026): Optimality
Anna L. Marchetti, Bo Zhang, Jan H. Novak, Amal Cherif, Multi Tier Cooperative Population Search for Continuous Black Box Optimization , Optimality: Vol. 3 No. 4 (2026): Optimality
Khris Benjamin Swain, Elif Ceviz, Maduabuchi Okafor, Rina V. Petrova, Hiroshi Tanabe, Stochastic Global Search on Molecular Potential-Energy Surfaces: A Narrative Survey and a Reported Numerical Illustration , Optimality: Vol. 3 No. 4 (2026): Optimality
Seyyed Farid Ghannadpour, Daniel A. Ferreira, Adaptive Evolutionary Search in Changing Environments: Moving Peaks and a Two-Species Chemostat Surrogate , Optimality: Vol. 3 No. 3 (2026): Optimality
Fatemeh Zare Baghiabad, Setayesh Zahiri, Zahra Joorbonyan, Cost-Normalized Constrained Surrogate Search for a Fed-Batch Model: Reported Sequential and Batch Comparisons , Optimality: Vol. 3 No. 3 (2026): Optimality
Paresh Kumar Panigrahi, Sukanta Nayak, An Inexact Line Search FQNDFP Algorithm to Investigate UncertainOptimization Problems , Optimality: Vol. 2 No. 2 (2025): OPT
Fatemeh Zahra Montazeri, From Conceptual Model to Empirical Validation: A Refined Decision Framework for Hybrid Simulation–Machine Learning–Optimization Architectures in Complex Systems , Optimality: Vol. 3 No. 2 (2026): Optimality

1-10 of 59 Next

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)

Hamiden Abd El-Wahed Khalifa, Darko Bozanic, Hashem Saberi Najafi, Pavan Kumar, Solving Vendor Selection Problem by Interval Approximation of Piecewise Quadratic Fuzzy Number , Optimality: Vol. 1 No. 1 (2024): OPT
Sheila Maria Muniz, Wagner Bandeira Andriola, Rita de Fátima Muniz, Seyyed Ahmad Edalatpanah, The Impacts of Covid-19 on Postgraduate Courses at the Federal University of Ceará (UFC, Brazil) , Optimality: Vol. 1 No. 1 (2024): OPT