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Abstract and Applied Analysis
Volume 2014, Article ID 192893, 6 pages
http://dx.doi.org/10.1155/2014/192893
Research Article

A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions

1School of Mathematics, Liaoning Normal University, Dalian 116029, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 12 January 2014; Accepted 20 March 2014; Published 14 April 2014

Academic Editor: Gaohang Yu

Copyright © 2014 Jie Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.