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January  2017, 13(1): 63-79. doi: 10.3934/jimo.2016004

## The stable duality of DC programs for composite convex functions

 1 School of Sciences, Zhejiang Agriculture and Forestry University, Hangzhou, Zhejiang 311300, China 2 Institute of Digital Media and Communication Technology, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China

* Corresponding author

Received  January 2015 Revised  June 2015 Published  March 2016

Fund Project: The work was supported by the Natural Science Foundation of China (11401533,11301484,11171247), the Scientific Research Foundation of Zhejiang Agriculture and Forestry University(2013FR080) and Nature science foundation of Zhejiang Province (LY14A010033).

In this paper, we consider a composite DC optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. By using the properties of the epigraph of the conjugate functions, some necessary and sufficient conditions which characterize the strong Fenchel-Lagrange duality and the stable strong Fenchel-Lagrange duality are given. We apply the results obtained to study the minmax optimization problem and $l_1$ penalty problem.

Citation: Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
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