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May  2020, 16(3): 1221-1233. doi: 10.3934/jimo.2018201

## Stability analysis for generalized semi-infinite optimization problems under functional perturbations

 Department of Mathematics, Bohai University, Jinzhou, Liaoning 121013, China

* Corresponding author: Xiaodong Fan (E-mail address: bhdxfxd@163.com)

Received  June 2017 Revised  October 2017 Published  December 2018

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 61572082), Natural Science Foundation of Liaoning Province of China (No. 20170540004, 20170540012) and Educational Commission of Liaoning Province of China (No. LZ2016003)

The concepts of essential solutions and essential solution sets for generalized semi-infinite optimization problems (GSIO for brevity) are introduced under functional perturbations, and the relations among the concepts of essential solutions, essential solution sets and lower semicontinuity of solution mappings are discussed. We show that a solution is essential if and only if the solution is unique; and a solution subset is essential if and only if it is the solution set itself. Some sufficient conditions for the upper semicontinuity of solution mappings are obtained. Finally, we show that every GSIO problem can be arbitrarily approximated by stable GSIO problems (the solution mapping is continuous), i.e., the set of all stable GSIO problems is dense in the set of all GSIO problems with the given topology.

Citation: Xiaodong Fan, Tian Qin. Stability analysis for generalized semi-infinite optimization problems under functional perturbations. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1221-1233. doi: 10.3934/jimo.2018201
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