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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.
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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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