2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications
| Format: | General Document |
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| _version_ | 1860798144679247872 |
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| building | INTELEK Repository |
| collection | Online Access |
| collectionurl | https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 |
| copyright | Copyright©PWB2025 |
| country | Malaysia |
| date | 2018-07-22 |
| format | General Document |
| id | 16164 |
| institution | UniSZA |
| originalfilename | A HYBRID BFGS-ZMRI METHOD FOR SOLVING UNCONSTRAINED OPTIMIZATION PROBLEMS AND ITS APPLICATIONS (PHD_2018).pdf |
| person | Zubai'ah Bt Zainal Abidin |
| recordtype | oai_dc |
| resourceurl | https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16164 |
| sourcemedia | Server storage Scanned document |
| spelling | 16164 https://intelek.unisza.edu.my/intelek/pages/view.php?ref=16164 https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection3 General Document Malaysia Library Staff (Top Management) Library Staff (Management) Library Staff (Support) Terengganu Faculty of Informatics & Computing English application/pdf 1.5 Server storage Scanned document Universiti Sultan Zainal Abidin UniSZA Private Access UNIVERSITI SULTAN ZAINAL ABIDIN SAMBox 2.3.4; modified using iTextSharp™ 5.5.10 ©2000-2016 iText Group NV (AGPL-version) 202 Copyright©PWB2025 2018-07-22 Mathematical optimization A HYBRID BFGS-ZMRI METHOD FOR SOLVING UNCONSTRAINED OPTIMIZATION PROBLEMS AND ITS APPLICATIONS (PHD_2018).pdf Zubai'ah Bt Zainal Abidin 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications The steepest descent (SD) method is known as one of the earliest method to minimize a function. The convergence rate of SD method is quite slow, but its simple form has made it the easiest method to be used especially in the form of computer codes. However, the SD method might not be efficient in terms of CPU times and convergence rate due to the zigzag phenomena occurred whenever the function is badly scaled. Therefore, many researchers tried to improve the performances of the SD method by introducing a new stepsize. Nevertheless, this newly developed approach seems to be very complicated and difficult to be implemented. In some conditions, the new stepsize methods lead to failure and sometimes do not retain the global convergence properties. Recently, Quasi-Newton method is extensively used in solving the problem of unconstrained optimization but due to its local convergent properties, this method may fail in solving nonconvex functions. In this research, a new search direction is introduced denoted as ZMRI (Zubai’ah, Mustafa, Rivaie and Ismail) method which is easy to use while retaining the global convergence properties. Then, a new hybrid search direction that combines search direction of quasi-Newton and ZMRI method which is known as hybrid BFGS-ZMRI method is proposed. The ZMRI method is compared with SD method whereas the hybrid BFGS-ZMRI method is compared with SD and BFGS method. These methods are tested with twenty nine standard optimization test problems by using MatlabR2011b. For every problem, five initial points are used; ranging from the point that is nearest to the solution point, to the one that is furthest away from it. The numerical results are based on the number of iterations and CPU time. The efficiency of the ZMRI and hybrid BFGS-ZMRI algorithms are analyzed by using performance profile. In addition, these new methods are also used to determine the function equation of real life situation based on data oil prices of RON95 in Malaysia. Theoretical proofs show that ZMRI and hybrid BFGS ZMRI methods fulfill sufficient descent conditions and possess global convergence properties. The numerical results show that ZMRI and hybrid BFGS-ZMRI outperformed SD method and BFGS. These new methods are also used to determine the function for oil prices of RON95 in Malaysia. Thus, we can conclude that the method ZMRI and hybrid BFGS-ZMRI managed to reduce the number of iterations and CPU time when compared to SD and BFGS method. The new method has been successfully define a function in real life based on data oil prices of RON95 in Malaysia. Dissertations, Academic BFGS Quasi-Newton Method Unconstrained Optimization Techniques Hybrid Numerical Optimization Method Thesis |
| spellingShingle | 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications |
| state | Terengganu |
| subject | Mathematical optimization Dissertations, Academic |
| summary | The steepest descent (SD) method is known as one of the earliest method to minimize a function. The convergence rate of SD method is quite slow, but its simple form has made it the easiest method to be used especially in the form of computer codes. However, the SD method might not be efficient in terms of CPU times and convergence rate due to the zigzag phenomena occurred whenever the function is badly scaled. Therefore, many researchers tried to improve the performances of the SD method by introducing a new stepsize. Nevertheless, this newly developed approach seems to be very complicated and difficult to be implemented. In some conditions, the new stepsize methods lead to failure and sometimes do not retain the global convergence properties. Recently, Quasi-Newton method is extensively used in solving the problem of unconstrained optimization but due to its local convergent properties, this method may fail in solving nonconvex functions. In this research, a new search direction is introduced denoted as ZMRI (Zubai’ah, Mustafa, Rivaie and Ismail) method which is easy to use while retaining the global convergence properties. Then, a new hybrid search direction that combines search direction of quasi-Newton and ZMRI method which is known as hybrid BFGS-ZMRI method is proposed. The ZMRI method is compared with SD method whereas the hybrid BFGS-ZMRI method is compared with SD and BFGS method. These methods are tested with twenty nine standard optimization test problems by using MatlabR2011b. For every problem, five initial points are used; ranging from the point that is nearest to the solution point, to the one that is furthest away from it. The numerical results are based on the number of iterations and CPU time. The efficiency of the ZMRI and hybrid BFGS-ZMRI algorithms are analyzed by using performance profile. In addition, these new methods are also used to determine the function equation of real life situation based on data oil prices of RON95 in Malaysia. Theoretical proofs show that ZMRI and hybrid BFGS ZMRI methods fulfill sufficient descent conditions and possess global convergence properties. The numerical results show that ZMRI and hybrid BFGS-ZMRI outperformed SD method and BFGS. These new methods are also used to determine the function for oil prices of RON95 in Malaysia. Thus, we can conclude that the method ZMRI and hybrid BFGS-ZMRI managed to reduce the number of iterations and CPU time when compared to SD and BFGS method. The new method has been successfully define a function in real life based on data oil prices of RON95 in Malaysia. |
| title | 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications |
| title_full | 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications |
| title_fullStr | 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications |
| title_full_unstemmed | 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications |
| title_short | 2018_A Hybrid BFGS-ZMRI Method for Solving Unconstrained Optimization Problems and Its Applications |
| title_sort | 2018_a hybrid bfgs-zmri method for solving unconstrained optimization problems and its applications |