Abstract
In this paper, we consider a set optimization problem with a partial order relation, which is defined by Minkowski difference. By using the image space analysis, we establish the relationships among the set optimization problem, a vector optimization problem and a set-valued optimization with vector criterion related to the image of the set optimization problem. In addition, two nonlinear regular weak separation functions are proposed for the set optimization problem. Based on the two nonlinear regular weak separation functions, saddle point sufficient optimality conditions, gap functions and error bounds for the set optimization problem, are obtained. Finally, we explore some applications of the obtained results to investigate robust multi-objective optimization problems and verify the validity of the results in shortest path problems with data uncertainty and multi-criteria traffic network equilibrium problems with interval-valued cost functions.
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Acknowledgements
The authors would like to express their sincere gratitude to Professor Jafar Zafarani and the anonymous reviewers for their constructive suggestions toward the improvement of this paper. This research was supported by the National Natural Science Foundation of China (Grant Numbers: 11801051, 11601437) and the Natural Science Foundation of Chongqing (Grant Number: cstc2019jcyj-msxmX0075).
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Xu, YD., Zhou, CL. & Zhu, SK. Image Space Analysis for Set Optimization Problems with Applications. J Optim Theory Appl 191, 311–343 (2021). https://doi.org/10.1007/s10957-021-01939-3
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DOI: https://doi.org/10.1007/s10957-021-01939-3