A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions
Under the condition that the values of the objective function and its subgradient are computed approximately, we introduce a cutting plane and level bundle method for minimizing nonsmooth nonconvex functions by combining cutting plane method with the ideas of proximity control and level constraint....
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/192893 |
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| author | Jie Shen Dan Li Li-Ping Pang |
| author_facet | Jie Shen Dan Li Li-Ping Pang |
| author_sort | Jie Shen |
| collection | DOAJ |
| description | Under the condition that the values of the objective function and its subgradient are computed approximately, we introduce a cutting plane and level bundle method for minimizing nonsmooth nonconvex functions by combining cutting plane method with the ideas of proximity control and level constraint. The proposed algorithm is based on the construction of both a lower and an upper polyhedral approximation model to the objective function and calculates new iteration points by solving a subproblem in which the model is employed not only in the objective function but also in the constraints. Compared with other proximal bundle methods, the new variant updates the lower bound of the optimal value, providing an additional useful stopping test based on the optimality gap. Another merit is that our algorithm makes a distinction between affine pieces that exhibit a convex or a concave behavior relative to the current iterate. Convergence to some kind of stationarity point is proved under some looser conditions. |
| format | Article |
| id | doaj-art-d8e26691c0274629ac15266432a72140 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-d8e26691c0274629ac15266432a721402025-02-03T05:53:54ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/192893192893A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex FunctionsJie Shen0Dan Li1Li-Ping Pang2School of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaUnder the condition that the values of the objective function and its subgradient are computed approximately, we introduce a cutting plane and level bundle method for minimizing nonsmooth nonconvex functions by combining cutting plane method with the ideas of proximity control and level constraint. The proposed algorithm is based on the construction of both a lower and an upper polyhedral approximation model to the objective function and calculates new iteration points by solving a subproblem in which the model is employed not only in the objective function but also in the constraints. Compared with other proximal bundle methods, the new variant updates the lower bound of the optimal value, providing an additional useful stopping test based on the optimality gap. Another merit is that our algorithm makes a distinction between affine pieces that exhibit a convex or a concave behavior relative to the current iterate. Convergence to some kind of stationarity point is proved under some looser conditions.http://dx.doi.org/10.1155/2014/192893 |
| spellingShingle | Jie Shen Dan Li Li-Ping Pang A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions Abstract and Applied Analysis |
| title | A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions |
| title_full | A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions |
| title_fullStr | A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions |
| title_full_unstemmed | A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions |
| title_short | A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions |
| title_sort | cutting plane and level stabilization bundle method with inexact data for minimizing nonsmooth nonconvex functions |
| url | http://dx.doi.org/10.1155/2014/192893 |
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