Abstract and Applied Analysis

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

Jie Shen, Dan Li, and Li-Ping Pang

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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.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 192893, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412278791

Digital Object Identifier
doi:10.1155/2014/192893

Mathematical Reviews number (MathSciNet)
MR3198161

Zentralblatt MATH identifier
07021909

Citation

Shen, Jie; Li, Dan; Pang, Li-Ping. A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 192893, 6 pages. doi:10.1155/2014/192893. https://projecteuclid.org/euclid.aaa/1412278791


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