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Article

Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections

by
Bothina El-Sobky
1,
Yousria Abo-Elnaga
2,
Abd Allah A. Mousa
3,* and
Mohamed A. El-Shorbagy
4,5
1
Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Qism Bab Sharqi 21568, Egypt
2
Department of Basic Science, Higher Technological Institute, Tenth of Ramadan City 44629, Egypt
3
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
4
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
5
Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(13), 1551; https://doi.org/10.3390/math9131551
Submission received: 9 May 2021 / Revised: 24 June 2021 / Accepted: 28 June 2021 / Published: 1 July 2021
(This article belongs to the Special Issue Advanced Analytical and Numerical Techniques for Technological Processes and Applications)
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Abstract

In this paper, a penalty method is used together with a barrier method to transform a constrained nonlinear programming problem into an unconstrained nonlinear programming problem. In the proposed approach, Newton’s method is applied to the barrier Karush–Kuhn–Tucker conditions. To ensure global convergence from any starting point, a trust-region globalization strategy is used. A global convergence theory of the penalty–barrier trust-region (PBTR) algorithm is studied under four standard assumptions. The PBTR has new features; it is simpler, has rapid convergerce, and is easy to implement. Numerical simulation was performed on some benchmark problems. The proposed algorithm was implemented to find the optimal design of a canal section for minimum water loss for a triangle cross-section application. The results are promising when compared with well-known algorithms.
Keywords: nonlinear programming problem; penalty method; barrier method; trust-region mechanism; global convergence nonlinear programming problem; penalty method; barrier method; trust-region mechanism; global convergence

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MDPI and ACS Style

El-Sobky, B.; Abo-Elnaga, Y.; Mousa, A.A.A.; El-Shorbagy, M.A. Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections. Mathematics 2021, 9, 1551. https://doi.org/10.3390/math9131551

AMA Style

El-Sobky B, Abo-Elnaga Y, Mousa AAA, El-Shorbagy MA. Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections. Mathematics. 2021; 9(13):1551. https://doi.org/10.3390/math9131551

Chicago/Turabian Style

El-Sobky, Bothina, Yousria Abo-Elnaga, Abd Allah A. Mousa, and Mohamed A. El-Shorbagy. 2021. "Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections" Mathematics 9, no. 13: 1551. https://doi.org/10.3390/math9131551

APA Style

El-Sobky, B., Abo-Elnaga, Y., Mousa, A. A. A., & El-Shorbagy, M. A. (2021). Trust-Region Based Penalty Barrier Algorithm for Constrained Nonlinear Programming Problems: An Application of Design of Minimum Cost Canal Sections. Mathematics, 9(13), 1551. https://doi.org/10.3390/math9131551

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