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Mathematical Problems in Engineering
Volume 2015, Article ID 215310, 9 pages
http://dx.doi.org/10.1155/2015/215310
Research Article

An Approximate Redistributed Proximal 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 6 January 2015; Revised 28 April 2015; Accepted 28 April 2015

Academic Editor: Anna Pandolfi

Copyright © 2015 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

We describe an extension of the redistributed technique form classical proximal bundle method to the inexact situation for minimizing nonsmooth nonconvex functions. The cutting-planes model we construct is not the approximation to the whole nonconvex function, but to the local convexification of the approximate objective function, and this kind of local convexification is modified dynamically in order to always yield nonnegative linearization errors. Since we only employ the approximate function values and approximate subgradients, theoretical convergence analysis shows that an approximate stationary point or some double approximate stationary point can be obtained under some mild conditions.