# American Institute of Mathematical Sciences

October  2017, 13(4): 1859-1881. doi: 10.3934/jimo.2017022

## Non-convex semi-infinite min-max optimization with noncompact sets

 1 School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China 2 School of Management Science, Qufu Normal University, Rizhao Shandong, 276826, China

* Corresponding author: Meixia Li

Received  October 2015 Revised  October 2016 Published  December 2016

Fund Project: This project is supported by the Natural Science Foundation of China (Grant No. 11571120,11271226,11271233,11401438,11401331), Shandong Provincial Natural Science Foundation (Grant No. ZR2013FL032) and the Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J14LI52).

In this paper, first we study the non-convex sup-type functions with noncompact sets. Under quite mild conditions, the expressions of its derivative and subderivative along arbitrary direction are given. Furthermore, the structure of its subdifferential is characterized completely. Then, using these results, we establish first-order optimality conditions for semi-infinite min-max optimization problems. These results generalize and improve the corresponding results in the relevant literatures.

Citation: Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022
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