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RESEARCH ARTICLE

A two-stage pareto selection strategy with compromise programming for the DIRECT algorithm

Mira Mustika1 ,  Salmah Salmah2* ,  Indarsih Indarsih2
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1 Department of Mathematics, Institut Teknologi Sumatera, Lampung , Indonesia
2 Department of Mathematics, Universitas Gadjah Mada, Yogyakarta , Indonesia
IJOCTA, 026260135 https://doi.org/10.36922/IJOCTA026260135
Received: 26 June 2026 | Revised: 25 August 2026 | Accepted: 1 September 2026 | Published online: 21 September 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

A novel DIRECT-type algorithm employing a parameterized Pareto approach is proposed for box-constrained optimization problems. The method features a two-stage Pareto selection process. The first stage focuses on global exploration, using a parameter to identify the smallest significant hyperrectangle, which helps avoid prolonged entrapment in local minima. The second stage emphasizes local exploitation to improve convergence speed. Instead of selecting all non-dominated hyperrectangles during this stage, the algorithm reduces then to three groups: smallest, medium, and largest. A key innovation is the introduction of the medium hyperrectangle selection procedure in this stage based on the Compromise Programming solution approach. In addition, the algorithm incorporates diagonal sampling and bisection procedures for hyperrectangles, with hyperrectangle size measured using the infinity norm rather than the Euclidean norm. The performance of the proposed algorithm is evaluated through numerical experiments on benchmark test problems and ten GKLS test classes (1000 test functions in total). Numerical results show that the proposed algorithm is competitive with, and generally outperforms, several existing DIRECT-type algorithms. Moreover, the results obtained onthe ten GKLS test classes further support the effectiveness of the proposed algorithm across all considered problem dimensions.

Keywords
Global optimization
DIRECT-type algorithm
Two-stage selection
Parameterized pareto approach
Compromise programming
Funding
None.
Conflict of interest
The authors declare they have no competing interests.
References
  1. Pardalos PM, Romeijn HE, Tuy H. Recent developments and trends in global optimization. J Comput Appl Math. 2000;124(1-2):209-228. https://doi.org/10.1016/S0377-0427(00)00425-8
  2. Alarie S, Audet C, Gheribi AE, Kokkolaras M, Le Digabel S. Two decades of blackbox optimization applications. EURO J Comput Optim. 2021;9:100011. https://doi.org/10.1016/j.ejco.2021.100011
  3. Chaudhary R, Verma P, Salgotra R, Gandomi AH. Five decades of genetic algorithms: A systematic and bibliometric review (1975-2025). Arch Comput Methods Eng. 2026;1-75. https://doi.org/10.1007/s11831-025-10487-2
  4. Benedetti R, Dickson MM, Espa G, Pantalone F, Piersimoni F. A simulated annealing-based algorithm for selecting balanced samples. Comput Stat. 2022;37(1):491-505. https://doi.org/10.1007/s00180-021-01113-3
  5. Okulewicz M, Zaborski M, Mańdziuk J. Self-adapting particle swarm optimization for continuous black box optimization. Appl Soft Comput. 2022;131:109722. https://doi.org/10.1016/j.asoc.2022.109722
  6. Shadkam E, Safari S, Abdollahzadeh SS. Finally, which meta-heuristic algorithm is the best one? Int J Decis Sci Risk Manag. 2021;10(1-2):32-50. https://doi.org/10.1504/IJDSRM.2021.117555
  7. Liberti L, Kucherenko S. Comparison of deterministic and stochastic approaches to global optimization. Int Trans Oper Res. 2005;12(3):263-285. https://doi.org/10.1111/j.1475-3995.2005.00503.x
  8. Vu KK, D'Ambrosio C, Hamadi Y, Liberti L. Surrogate-based methods for black-box optimization. Int Trans Oper Res. 2017;24(3):393-424. https://doi.org/10.1111/itor.12292
  9. Liang J, Lou Y, Yu M, Bi Y, Yu K. A survey of surrogate-assisted evolutionary algorithms for expensive optimization. J Membr Comput. 2025;7(2):108-127. https://doi.org/10.1007/s41965-024-00165-w
  10. Jones DR, Perttunen CD, Stuckman BE. Lipschitzian optimization without the Lipschitz constant. J Optim Theory Appl. 1993;79(1):157-181. https://doi.org/10.1007/BF00941892
  11. Stripinis L, Paulavičius R. Derivative-Free DIRECT-Type Global Optimization: Applications and Software. Cham, Switzerland: Springer; 2023. https://doi.org/10.1007/978-3-031-46537-6
  12. Li C, Chen Y, Yang X, Wang Z, Lu Z, Chi X. Intelligent Black-Litterman portfolio optimization using a decomposition-based multi-objective DIRECT algorithm. Appl Sci. 2022;12(14):7089. https://doi.org/10.3390/app12147089
  13. Rousseau A, Pagerit S, Gao DW. Plug-in hybrid electric vehicle control strategy parameter optimization. J Asian Electr Veh. 2008;6(2):1125-1133. https://doi.org/10.4130/jaev.6.1125
  14. Panday A, Bansal HO. Fuel efficiency optimization of input-split hybrid electric vehicle using DIRECT algorithm. In: 2014 9th International Conference on Industrial and Information Systems (ICIIS). IEEE; 2014:1-6. https://doi.org/10.1109/ICIINFS.2014.7036640
  15. Barmuta P, Mercuri M, Soh PJ, Karsmakers P, Vandenbosch GAE, Leroux P, et al. Radar range improvement using gradient-free optimization for health care applications. In: 2016 21st International Conference on Microwave, Radar and Wireless Communications (MIKON). IEEE; 2016:1-4. https://doi.org/10.1109/MIKON.2016.7491968
  16. Cong S, Wang YH, Cheng JY. Coal mine microseismic velocity model inversion based on first arrival time difference. Arab J Geosci. 2019;12(1):5. https://doi.org/10.1007/s12517-018-4172-4
  17. Larsson E, Simonsen MH, Mao W. DIRECT optimization algorithm in weather routing of ships. In: Proceedings of the International Offshore and Polar Engineering Conference. ISOPE; 2015:1207-1214. Accessed October 28, 2025. https://onepetro.org/ISOPEIOPEC/proceedings-abstract/ISOPE15/ISOPE15/ISOPE-I-15-170/14504
  18. Hao J, Yu Z, Zhao Z, Shen P, Zhan X. Optimization of key parameters of energy management strategy for hybrid electric vehicle using DIRECT algorithm. Energies. 2016;9(12):997. https://doi.org/10.3390/en9120997
  19. Wang N, Tsai CM, Cha KC. A study of parallel efficiency of modified DIRECT algorithm applied to thermohydrodynamic lubrication. J Mech. 2009;25(2):143-150. https://doi.org/10.1017/S1727719100002598
  20. Zheng C, Calvin J, Gotsman C. A DIRECT-type global optimization algorithm for image registration. J Glob Optim. 2021;79(2):431-445. https://doi.org/10.1007/s10898-020-00914-y
  21. Campana EF, Diez M, Liuzzi G, Lucidi S, Pellegrini R, Piccialli V, et al. A multi-objective DIRECT algorithm for ship hull optimization. Comput Optim Appl. 2018;71(1):53-72. https://doi.org/10.1007/s10589-017-9955-0
  22. Wachowiak MP, Peters TM. High-performance medical image registration using new optimization techniques. IEEE Trans Inf Technol Biomed. 2006;10(2):344-353. https://doi.org/10.1109/TITB.2006.864476
  23. Scitovski R, Sabo K. Application of the DIRECT algorithm to searching for an optimal k-partition of the set A ⊂ R^n and its application to the multiple circle detection problem. J Glob Optim. 2019;74(1):63-77. https://doi.org/10.1007/s10898-019-00743-8
  24. Shen J, Dusmez S, Khaligh A. Optimization of sizing and battery cycle life in battery/ultracapacitor hybrid energy storage systems for electric vehicle applications. IEEE Trans Ind Inform. 2014;10(4):2112-2121. https://doi.org/10.1109/TII.2014.2334233
  25. Liuzzi G, Lucidi S, Piccialli V. Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization. Comput Optim Appl. 2016;65(2):449-475. https://doi.org/10.1007/s10589-015-9741-9
  26. Liu Q, Yang G, Zhang Z, Zeng J. Improving the convergence rate of the DIRECT global optimization algorithm. J Glob Optim. 2017;67(4):851-872. https://doi.org/10.1007/s10898-016-0447-z
  27. Paulavičius R, Sergeyev YD, Kvasov DE, Žilinskas J. Globally-biased BIRECT algorithm with local accelerators for expensive global optimization. Expert Syst Appl. 2020;144:113052. https://doi.org/10.1016/j.eswa.2019.113052
  28. Paulavičius R, Chiter L, Žilinskas J. Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants. J Glob Optim. 2018;71(1):5-20. https://doi.org/10.1007/s10898-016-0485-6
  29. Sergeyev YD, Kvasov DE. Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J Optim. 2006;16(3):910-937. https://doi.org/10.1137/040621132
  30. Guessoum N, Chiter L. Diagonal partitioning strategy using bisection of rectangles and a novel sampling scheme. MENDEL. 2023;29(2):131-146. https://doi.org/10.13164/mendel.2023.2.131
  31. Stripinis L, Paulavičius R. Lipschitz-inspired HALRECT algorithm for derivative-free global optimization. J Glob Optim. 2024;88(1):139-169. https://doi.org/10.1007/s10898-023-01296-7
  32. Jones DR. The DIRECT global optimization algorithm. In: Floudas CA, Pardalos PM, eds. Encyclopedia of Optimization. Boston, MA: Springer; 2001:431-440. https://doi.org/10.1007/0-306-48332-7_93
  33. Mockus J. On the Pareto optimality in the context of Lipschitzian optimization. Informatica. 2011;22(4):521-536. https://doi.org/10.15388/Informatica.2011.340
  34. Mockus J, Paulavičius R. On the reduced-set Pareto-Lipschitzian optimization. Comput Sci Tech. 2013;1(2):184-192. https://doi.org/10.15181/csat.v1i2.84
  35. Stripinis L, Paulavičius R, Žilinskas J. Improved scheme for selection of potentially optimal hyper-rectangles in DIRECT. Optim Lett. 2018;12(7):1699-1712. https://doi.org/10.1007/s11590-017-1228-4
  36. Mustika M, Salmah, Indarsih. A new DIRECT-type algorithm based on bisection of rectangles and diagonal sampling with the Pareto approach. IAENG Int J Appl Math. 2024;54(7):1352-1361. Accessed August 18, 2025. https://www.iaeng.org/IJAM/issues_v54/issue_7/IJAM_54_7_14.pdf
  37. Stripinis L, Paulavičius R. DIRECTGO: A new DIRECT-type MATLAB toolbox for derivative-free global optimization. ACM Trans Math Softw. 2022;48(4):1-46. https://doi.org/10.1145/3559755
  38. Stripinis L, Paulavičius R. An empirical study of various candidate selection and partitioning techniques in the DIRECT framework. J Glob Optim. 2024;88(3):723-753. https://doi.org/10.1007/s10898-022-01185-5
  39. Jones DR, Martins JRRA. The DIRECT algorithm: 25 years later. J Glob Optim. 2021;79(3):521-566. https://doi.org/10.1007/s10898-020-00952-6
  40. Mockus J, Paulavičius R, Rusakevičius D, Šešok D, Žilinskas J. Application of reduced-set Pareto-Lipschitzian optimization to truss optimization. J Glob Optim. 2017;67(1-2):425-450. https://doi.org/10.1007/s10898-015-0364-6
  41. Zeleny M. A concept of compromise solutions and the method of the displaced ideal. Comput Oper Res. 1974;1(3-4):479-496. https://doi.org/10.1016/0305-0548(74)90064-1
  42. Romero C, Rehman T. Chapter five: Compromise programming. In: Multiple Criteria Analysis for Agricultural Decisions. Amsterdam, Netherlands: Elsevier; 2003:63-78. https://doi.org/10.1016/S0926-5589(03)80007-9
  43. Hedar A. Test Functions for Unconstrained Global Optimization [online]. 2005. Accessed July 23, 2023. http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO.htm
  44. Molga M, Smutnicki C. Test Functions for Optimization Needs [online]. Wrocław, Poland: Wrocław University of Technology; 2005. Accessed July 9, 2024. https://robertmarks.org/Classes/ENGR5358/Papers/functions.pdf
  45. Gaviano M, Kvasov DE, Lera D, Sergeyev YD. Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans Math Softw. 2003;29(4):469-480. https://doi.org/10.1145/962437.962444
  46. Lera D, Sergeyev YD. GOSH: Derivative-free global optimization using multi-dimensional space-filling curves. J Glob Optim. 2018;71(1):193-211. https://doi.org/10.1007/s10898-017-0589-7
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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