Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems
To solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invarian...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/324181 |
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| author | Chein-Shan Liu |
| author_facet | Chein-Shan Liu |
| author_sort | Chein-Shan Liu |
| collection | DOAJ |
| description | To solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invariant manifold defined for the model problem in terms of a locally quadratic function is used to derive a purely iterative algorithm and the convergence is proven. Then, the rank-two updating techniques of BFGS are employed, which result in several novel algorithms as being faster than the steepest descent method (SDM) and the variable metric method (DFP). Six numerical examples are examined and compared with exact solutions, revealing that the new algorithms of OA, GOA, and the updated ones have superior computational efficiency and accuracy. |
| format | Article |
| id | doaj-art-3d6fbcd61699436ebc11d237380b6826 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-3d6fbcd61699436ebc11d237380b68262025-02-03T01:31:34ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/324181324181Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization ProblemsChein-Shan Liu0Department of Civil Engineering, National Taiwan University, Taipei 106-17, TaiwanTo solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invariant manifold defined for the model problem in terms of a locally quadratic function is used to derive a purely iterative algorithm and the convergence is proven. Then, the rank-two updating techniques of BFGS are employed, which result in several novel algorithms as being faster than the steepest descent method (SDM) and the variable metric method (DFP). Six numerical examples are examined and compared with exact solutions, revealing that the new algorithms of OA, GOA, and the updated ones have superior computational efficiency and accuracy.http://dx.doi.org/10.1155/2014/324181 |
| spellingShingle | Chein-Shan Liu Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems Journal of Applied Mathematics |
| title | Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems |
| title_full | Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems |
| title_fullStr | Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems |
| title_full_unstemmed | Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems |
| title_short | Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems |
| title_sort | optimal algorithms and the bfgs updating techniques for solving unconstrained nonlinear minimization problems |
| url | http://dx.doi.org/10.1155/2014/324181 |
| work_keys_str_mv | AT cheinshanliu optimalalgorithmsandthebfgsupdatingtechniquesforsolvingunconstrainednonlinearminimizationproblems |