{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T14:24:17Z","timestamp":1784730257707,"version":"3.55.0"},"reference-count":58,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,4]],"date-time":"2026-01-04T00:00:00Z","timestamp":1767484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This paper proposes JADEGBO, a hybrid gradient-based metaheuristic for solving complex single- and multi-constraint engineering design problems as well as cost-sensitive security optimisation tasks. The method combines Adaptive Differential Evolution with Optional External Archive (JADE), which provides self-adaptive exploration through p-best mutation, an external archive, and success-based parameter learning, with the Gradient-Based Optimiser (GBO), which contributes Newton-inspired gradient search rules and a local escaping operator. In the proposed scheme, JADE is first employed to discover promising regions of the search space, after which GBO performs an intensified local refinement of the best individuals inherited from JADE. The performance of JADEGBO is assessed on the CEC2017 single-objective benchmark suite and compared against a broad set of classical and recent metaheuristics. Statistical indicators, convergence curves, box plots, histograms, sensitivity analyses, and scatter plots show that the hybrid typically attains the best or near-best mean fitness, exhibits low run-to-run variance, and maintains a favourable balance between exploration and exploitation across rotated, shifted, and composite landscapes. To demonstrate practical relevance, JADEGBO is further applied to the following four well-known constrained engineering design problems: welded beam, pressure vessel, speed reducer, and three-bar truss design. The algorithm consistently produces feasible high-quality designs and closely matches or improves upon the best reported results while keeping computation time competitive.<\/jats:p>","DOI":"10.3390\/computation14010011","type":"journal-article","created":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T08:40:53Z","timestamp":1767602453000},"page":"11","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Hybrid Gradient-Based Optimiser for Solving Complex Engineering Design Problems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9060-7188","authenticated-orcid":false,"given":"Jamal","family":"Zraqou","sequence":"first","affiliation":[{"name":"Department of Computer Science, Faculty of Information Technology, University of Petra, Amman 11196, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Riyad","family":"Alrousan","sequence":"additional","affiliation":[{"name":"Design and Visual Communication Department, School of Architecture and Built Environment (SABE), German Jordanian University (GJU), Amman 11180, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-2013-6923","authenticated-orcid":false,"given":"Zaid","family":"Khrisat","sequence":"additional","affiliation":[{"name":"Faculty of Arts and Educational Sciences, Middle East University, Amman 11831, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7548-0529","authenticated-orcid":false,"given":"Faten","family":"Hamad","sequence":"additional","affiliation":[{"name":"Information Science and Technology Department, Sultan Qaboos University, Muscat, Oman"},{"name":"Information Science Department, School of Educational Sciences, The University of Jordan, Amman 11942, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-5328-7812","authenticated-orcid":false,"given":"Niveen","family":"Halalsheh","sequence":"additional","affiliation":[{"name":"Department of Journalism, Media, and Digital Communication, School of Arts, The University of Jordan, Amman 11942, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9170-3291","authenticated-orcid":false,"given":"Hussam","family":"Fakhouri","sequence":"additional","affiliation":[{"name":"Faculty of Artificial Intelligence, Al-Balqa Applied University, Al-Salt 19117, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1016\/j.cirp.2012.05.001","article-title":"Complexity in engineering design and manufacturing","volume":"61","author":"ElMaraghy","year":"2012","journal-title":"CIRP Ann."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chau, M., Fu, M.C., Qu, H., and Ryzhov, I.O. (2014, January 7\u201310). Simulation optimization: A tutorial overview and recent developments in gradient-based methods. Proceedings of the Winter Simulation Conference 2014, Savannah, GA, USA.","DOI":"10.1109\/WSC.2014.7019875"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1080\/00207721.2013.823526","article-title":"An overview of population-based algorithms for multi-objective optimisation","volume":"46","author":"Giagkiozis","year":"2015","journal-title":"Int. J. Syst. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1145\/291889.291893","article-title":"The design, implementation, and evaluation of Jade","volume":"20","author":"Rinard","year":"1998","journal-title":"ACM Trans. Program. Lang. Syst. (TOPLAS)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1109\/TEVC.2009.2014613","article-title":"JADE: Adaptive differential evolution with optional external archive","volume":"13","author":"Zhang","year":"2009","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.ins.2020.06.037","article-title":"Gradient-based optimizer: A new metaheuristic optimization algorithm","volume":"540","author":"Ahmadianfar","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2431","DOI":"10.1007\/s11831-022-09872-y","article-title":"Gradient-based optimizer (gbo): A review, theory, variants, and applications","volume":"30","author":"Daoud","year":"2023","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_8","first-page":"11","article-title":"A new hybrid matheuristic optimization algorithm for solving design and network engineering problems","volume":"13","author":"Chagwiza","year":"2018","journal-title":"Int. J. Manag. Sci. Eng. Manag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5316379","DOI":"10.1155\/2018\/5316379","article-title":"A Novel Hybrid Algorithm for Solving Multiobjective Optimization Problems with Engineering Applications","volume":"2018","author":"Fan","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.engappai.2019.06.017","article-title":"A hybrid optimization algorithm based on cuckoo search and differential evolution for solving constrained engineering problems","volume":"85","author":"Zhang","year":"2019","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_11","unstructured":"Yassin, B., Lahcen, A., and Es-Sadek, M.Z. (2019, January 25\u201326). A Hybrid Optimization Algorithm for Solving Constrained Engineering Design Problems. Proceedings of the 2019 5th International Conference on Optimization and Applications (ICOA), Kenitra, Morocco."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1699","DOI":"10.1007\/s00500-017-2894-y","article-title":"An efficient hybrid algorithm based on Water Cycle and Moth-Flame Optimization algorithms for solving numerical and constrained engineering optimization problems","volume":"23","author":"Khalilpourazari","year":"2019","journal-title":"Soft Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"9404","DOI":"10.1007\/s11227-020-03212-2","article-title":"Efficient hybrid algorithm based on moth search and fireworks algorithm for solving numerical and constrained engineering optimization problems","volume":"76","author":"Han","year":"2020","journal-title":"J. Supercomput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3091","DOI":"10.1007\/s13369-019-04285-9","article-title":"Hybrid Particle Swarm Optimization with Sine Cosine Algorithm and Nelder\u2013Mead Simplex for Solving Engineering Design Problems","volume":"45","author":"Fakhouri","year":"2020","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wang, S., Jia, H., Abualigah, L.M.Q., Liu, Q., and Zheng, R. (2021). An improved hybrid aquila optimizer and harris hawks algorithm for solving industrial engineering optimization problems. Processes, 9.","DOI":"10.3390\/pr9091551"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1515\/mt-2020-0093","article-title":"A novel hybrid water wave optimization algorithm for solving complex constrained engineering problems","volume":"63","author":"Pholdee","year":"2021","journal-title":"Mater. Test."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Brajevic, I., Stanimirovi\u0107, P.S., Li, S., Cao, X., Khan, A.T., and Kazakovtsev, L.A. (2022). Hybrid Sine Cosine Algorithm for Solving Engineering Optimization Problems. Mathematics, 10.","DOI":"10.3390\/math10234555"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s40430-022-03700-x","article-title":"Hybrid teaching\u2013learning-based optimization for solving engineering and mathematical problems","volume":"44","author":"Dastan","year":"2022","journal-title":"J. Braz. Soc. Mech. Sci. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Liu, Q., Li, N., Jia, H., Qi, Q., Abualigah, L.M.Q., and Liu, Y. (2022). A Hybrid Arithmetic Optimization and Golden Sine Algorithm for Solving Industrial Engineering Design Problems. Mathematics, 10.","DOI":"10.3390\/math10091567"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zhang, M., Wang, D., and Yang, J. (2022). Hybrid-Flash Butterfly Optimization Algorithm with Logistic Mapping for Solving the Engineering Constrained Optimization Problems. Entropy, 24.","DOI":"10.3390\/e24040525"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8357","DOI":"10.1038\/s41598-024-59034-2","article-title":"A hybrid particle swarm optimization algorithm for solving engineering problem","volume":"14","author":"Qiao","year":"2024","journal-title":"Sci. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1007\/s44196-024-00439-2","article-title":"Hybrid Strategies Based Seagull Optimization Algorithm for Solving Engineering Design Problems","volume":"17","author":"Hou","year":"2024","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhang, Y., and Xing, L. (2024). A New Hybrid Improved Arithmetic Optimization Algorithm for Solving Global and Engineering Optimization Problems. Mathematics, 12.","DOI":"10.3390\/math12203221"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Tang, W., Cao, L., Chen, Y., Chen, B., and Yue, Y. (2024). Solving Engineering Optimization Problems Based on Multi-Strategy Particle Swarm Optimization Hybrid Dandelion Optimization Algorithm. Biomimetics, 9.","DOI":"10.3390\/biomimetics9050298"},{"key":"ref_25","first-page":"219","article-title":"BHJO: A Novel Hybrid Metaheuristic Algorithm Combining the Beluga Whale, Honey Badger, and Jellyfish Search Optimizers for Solving Engineering Design Problems","volume":"141","author":"Zitouni","year":"2024","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Singh, G., Biswas, S., Maiti, B., and Kumar Bera, U.K. (2024, January 15\u201317). A Novel Hybrid Gazelle Optimization Algorithm with Differential Evolution for Solving Engineering Design Problem. Proceedings of the 2024 IEEE Silchar Subsection Conference (SILCON 2024), Agartala, India.","DOI":"10.1109\/SILCON63976.2024.10910408"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Al Hwaitat, A.K., Fakhouri, H.N., Zraqou, J.S., and Sirhan, N.N. (2025). Hybrid Optimization Algorithm for Solving Attack-Response Optimization and Engineering Design Problems. Algorithms, 18.","DOI":"10.3390\/a18030160"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1007\/s13198-024-02609-z","article-title":"A novel hybrid ESO-DE-WHO algorithm for solving real-engineering optimization problems","volume":"16","author":"Panigrahy","year":"2025","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Jonnalagadda, A.K., Acharya, S., Pasam, V.R., and Shahane, R. (2025, January 20\u201322). Quantum-AI Hybrid Algorithms for Solving Large-Scale Engineering Problems Using LLM-Driven Optimizations. Proceedings of the 2025 5th International Conference on Intelligent Technologies (CONIT), Hubbali, India.","DOI":"10.1109\/CONIT65521.2025.11166726"},{"key":"ref_30","first-page":"967","article-title":"MOCBOA: Multi-Objective Chef-Based Optimization Algorithm Using Hybrid Dominance Relations for Solving Engineering Design Problems","volume":"143","author":"Chalabi","year":"2025","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s40430-025-05697-5","article-title":"Advancements and emerging trends in integrating machine learning and deep learning for SHM in mechanical and civil engineering: A comprehensive review","volume":"47","author":"Khatir","year":"2025","journal-title":"J. Braz. Soc. Mech. Sci. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4958","DOI":"10.1038\/s41598-022-09126-8","article-title":"Damage assessment in structures using artificial neural network working and a hybrid stochastic optimization","volume":"12","author":"Khatir","year":"2022","journal-title":"Sci. Rep."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1038\/s44172-025-00431-4","article-title":"Data-intelligence driven methods for durability, damage diagnosis and performance prediction of concrete structures","volume":"4","author":"Li","year":"2025","journal-title":"Commun. Eng."},{"key":"ref_34","first-page":"79","article-title":"Robust Email Spam Filtering Using a Hybrid of Grey Wolf Optimiser and Naive Bayes Classifier","volume":"23","author":"Zraqou","year":"2023","journal-title":"Cybern. Inf. Technol."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Zraqou, J., Alkhadour, W., and Hadi, W. (2024, January 26\u201328). Vegetation Change Detection in Amman, Jordan Using Remote Sensing and GIS. Proceedings of the 2024 2nd International Conference on Cyber Resilience (ICCR), Dubai, United Arab Emirates.","DOI":"10.1109\/ICCR61006.2024.10533083"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhang, L.M., Dahlmann, C., and Zhang, Y. (2009, January 20\u201322). Human-inspired Algorithms for Continuous Function Optimization. Proceedings of the 2009 IEEE International Conference on Intelligent Computing and Intelligent Systems, Shanghai, China.","DOI":"10.1109\/ICICISYS.2009.5357838"},{"key":"ref_37","unstructured":"Algorithm Afternoon (2025, December 06). Adaptive Differential Evolution with Optional External Archive. Available online: https:\/\/algorithmafternoon.com\/differential\/adaptive_differential_evolution_with_optional_external_archive\/."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"107250","DOI":"10.1016\/j.cie.2021.107250","article-title":"Aquila Optimizer: A novel meta-heuristic optimization algorithm","volume":"157","author":"Abualigah","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"113338","DOI":"10.1016\/j.eswa.2020.113338","article-title":"Chimp optimization algorithm","volume":"149","author":"Khishe","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_40","first-page":"e01995","article-title":"The past, present and future of the pangolin in Mainland China","volume":"33","author":"Zhang","year":"2022","journal-title":"Glob. Ecol. Conserv."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2571863","DOI":"10.1155\/2021\/2571863","article-title":"Dingo optimizer: A nature-inspired metaheuristic approach for engineering problems","volume":"2021","author":"Bairwa","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Falahah, I.A., Al-Baik, O., Alomari, S., Bektemyssova, G., Gochhait, S., Leonova, I., Malik, O.P., Werner, F., and Dehghani, M. (2024). Frilled Lizard Optimization: A Novel Nature-Inspired Metaheuristic Algorithm for Solving Optimization Problems. Preprints, 2024030898.","DOI":"10.20944\/preprints202403.0898.v1"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Jahn, J. (2009). Vector Optimization, Springer.","DOI":"10.1007\/978-3-642-17005-8_9"},{"key":"ref_44","unstructured":"Mathew, T.V. (2012). Genetic Algorithm, IIT Bombay. submitted."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey wolf optimizer","volume":"69","author":"Mirjalili","year":"2014","journal-title":"Adv. Eng. Softw."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.knosys.2015.07.006","article-title":"Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm","volume":"89","author":"Mirjalili","year":"2015","journal-title":"Knowl. Based Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/s00521-015-1870-7","article-title":"Multi-verse optimizer: A nature-inspired algorithm for global optimization","volume":"27","author":"Mirjalili","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s00500-016-2474-6","article-title":"Particle swarm optimization algorithm: An overview","volume":"22","author":"Wang","year":"2018","journal-title":"Soft Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"115665","DOI":"10.1016\/j.eswa.2021.115665","article-title":"Remora optimization algorithm","volume":"185","author":"Jia","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Nikolaev, A.G., and Jacobson, S.H. (2010). Simulated annealing. Handbook of Metaheuristics, Springer.","DOI":"10.1007\/978-1-4419-1665-5_1"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.knosys.2015.12.022","article-title":"SCA: A sine cosine algorithm for solving optimization problems","volume":"96","author":"Mirjalili","year":"2016","journal-title":"Knowl.-Based Syst."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Wu, D., Rao, H., Wen, C., Jia, H., Liu, Q., and Abualigah, L. (2022). Modified sand cat swarm optimization algorithm for solving constrained engineering optimization problems. Mathematics, 10.","DOI":"10.3390\/math10224350"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"6461","DOI":"10.1007\/s11227-021-04093-9","article-title":"Success history intelligent optimizer","volume":"78","author":"Fakhouri","year":"2022","journal-title":"J. Supercomput."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.knosys.2018.11.024","article-title":"Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems","volume":"165","author":"Dhiman","year":"2019","journal-title":"Knowl.-Based Syst."},{"key":"ref_55","first-page":"2557","article-title":"Synergistic swarm optimization algorithm","volume":"139","author":"Alzoubi","year":"2024","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Singh, A., Sharma, A., Rajput, S., Mondal, A.K., Bose, A., and Ram, M. (2022). Parameter extraction of solar module using the sooty tern optimization algorithm. Electronics, 11.","DOI":"10.3390\/electronics11040564"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The whale optimization algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"5211","DOI":"10.1038\/s41598-023-31876-2","article-title":"American zebra optimization algorithm for global optimization problems","volume":"13","author":"Mohapatra","year":"2023","journal-title":"Sci. Rep."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/14\/1\/11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T05:29:19Z","timestamp":1767763759000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/14\/1\/11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,4]]},"references-count":58,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["computation14010011"],"URL":"https:\/\/doi.org\/10.3390\/computation14010011","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,4]]}}}