On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions
We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of comp...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/620949 |
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| _version_ | 1832558542685470720 |
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| author | Bingzhuang Liu Changyu Wang Wenling Zhao |
| author_facet | Bingzhuang Liu Changyu Wang Wenling Zhao |
| author_sort | Bingzhuang Liu |
| collection | DOAJ |
| description | We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem. |
| format | Article |
| id | doaj-art-9328e0fdeee044668c12af668dad7dc3 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-9328e0fdeee044668c12af668dad7dc32025-02-03T01:32:09ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/620949620949On the Convergence of a Smooth Penalty Algorithm without Computing Global SolutionsBingzhuang Liu0Changyu Wang1Wenling Zhao2School of Science, Shandong University of Technology, Zibo 255049, ChinaInstitute of Operations Research, Qufu Normal University, Qufu 273165, ChinaSchool of Science, Shandong University of Technology, Zibo 255049, ChinaWe consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem.http://dx.doi.org/10.1155/2012/620949 |
| spellingShingle | Bingzhuang Liu Changyu Wang Wenling Zhao On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions Journal of Applied Mathematics |
| title | On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions |
| title_full | On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions |
| title_fullStr | On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions |
| title_full_unstemmed | On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions |
| title_short | On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions |
| title_sort | on the convergence of a smooth penalty algorithm without computing global solutions |
| url | http://dx.doi.org/10.1155/2012/620949 |
| work_keys_str_mv | AT bingzhuangliu ontheconvergenceofasmoothpenaltyalgorithmwithoutcomputingglobalsolutions AT changyuwang ontheconvergenceofasmoothpenaltyalgorithmwithoutcomputingglobalsolutions AT wenlingzhao ontheconvergenceofasmoothpenaltyalgorithmwithoutcomputingglobalsolutions |