NONE 20260429120211038 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Tareq Hamadneh Department of Mathematics, Al Zaytoonah University of Jordan, Amman, JORDAN This paper presents global computational problems related to establishing lower bounds for highdimensional continuous functions over a simplex. We propose a rigorous approach to constructing affine lower bounds (ALB) that remain closely beneath the original function. By extending the least squares method for control Bernstein, we develop a convergent AB to both polynomials and rational functions. Furthermore, we demonstrate that these bounds achieve high convergence rates to the original functions. Finally, we evaluate our approach by comparing it with previous methods using error bound approximation. 04 29 2026 04 29 2026 51 6 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2026.25.6 https://wseas.com/journals/mathematics/2026/a125106-003(2026).pdf Claire S. Adjiman and Christodoulos A. Floudas. Rigorous convex underestimators for general twice-differentiable problems. Journal of Global Optimization, 9(1):23–40, 1996. https://doi.org/10.1007/BF00121749 doi:10.1007/BF00121749. Christodoulos A. Floudas. Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, New York, 1995. URL: https://doi.org/10.1093/oso/ 9780195100563.001.0001. Christian Jansson. Convex-concave extensions. BIT Numerical Mathematics, 40(2):291–313, 2000. https://doi.org/10.1023/A:1022343007844 doi:10.1023/A:1022343007844. Jurgen Garloff and Andrew P. Smith. A ¨ comparison of methods for the computation of affine lower bound functions for polynomials. In Christophe Jermann, Arnold Neumaier, and Djamila Sam, editors, Global Optimization and Constraint Satisfaction, volume 3478 of Lecture Notes in Computer Science, pages 71–85. Springer, Berlin, Heidel berg, 2005. https:// doi.org/10.1007/11425076_6. Faiz . A. Al-Khayyal and James E. Falk. Jointly constrained biconvex programming. Mathematics of Operations Research, 8(2):273–286, 1983. https://doi.org/10.1287/moor.8.2.273 doi:10.1287/moor.8.2.273. Jordan Ninin, Fr´ed´eric Messine, and Pierre Hansen. A reliable affine relaxation method for global optimization. 4OR: A Quarterly Journal of Operations Research, 13(3):247– 277, 2015. https://doi.org/10.1007/ s10288-014-0269-0 doi:10.1007/ s10288-014-0269-0. Shaojian Qu, Kecun Zhang, and Fusheng Wang. A global optimization using linear relaxation for generalized geometric programming. European Journal of Operational Research, 190(2):345–356, 2008. https://doi.org/10.1016/j.ejor.2007.06.034 doi:10.1016/j.ejor.2007.06.034. Andrew P Smith. Enclosure methods for systems of polynomial equations and inequalities. PhD thesis, Universit¨at Konstanz, konstanz, 2012. https://kops.uni-konstanz.de/ entities/publication/d024b254-f338-47debd1f-3b4fdbd944b4 Rida T. Farouki and V. T. Rajan. Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design, 5(1):1–26, 1988. https://doi.org/10.1016/0167-8396(88)90016-7 doi:10.1016/0167-8396(88)90016-7. Tareq Hamadneh, Mohammed Ali, and Hassan Al-Zoubi. Linear optimization of polynomial rational functions: Applications for positivity analysis. Mathematics, 8(2):283, 2020. https://doi.org/10.3390/math8020283 doi:10.3390/math8020283. Anthony Narkawicz, Jurgen ¨ Garloff, Andrew P. Smith, and C´esar A. Munoz. Bounding the range of a rational function over a box. Reliable Computing, 17(1):34–39, 2012. Haitham Qawaqneh, Hassan Jari, Ali Altalbe, and Ahmet Bekir. Stability analysis, modulation instability, and the analytical wave solitons to the fractional Boussinesq-Burgers system. Physica Scripta, 99(12):125235, 2024. https:// doi.org/10.1088/1402-4896/ad8e07 doi:10.1088/1402-4896/ad8e07. Richard Leroy. Convergence under subdivision and complexity of polynomial minimization in the simplicial bernstein basis. Reliable Computing, 17(1):11–21, 2012. https:// api.semanticscholar.org/CorpusID:11321842. Accessed: February 20, 2026. (text in French) Richard Leroy. Certificats de positivit´e et min-imisation polynomiale dans la base de Bernstein multivari´ee. PhD thesis, Universit´e Rennes 1, Rennes, France, 2008. Accessed: February 20, 2026. URL: https:// theses.hal.science/tel-00349444. Ying Ji, Huanhuan Li, and Huijie Zhang. Risk-averse two-stage stochastic minimum cost consensus models with asymmetric adjustment cost. Group Decision and Negotiation, 31(2):261–291, 2022. https://doi.org/10.1007/s10726-021-09752-z doi:10.1007/s10726-021-09752-z.