Abstract
In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant Numbers: 10871216 and 11171362). The authors thank the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.
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Communicated by Jafar Zafarani.
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Li, S.J., Xu, Y.D. & Zhu, S.K. Nonlinear Separation Approach to Constrained Extremum Problems. J Optim Theory Appl 154, 842–856 (2012). https://doi.org/10.1007/s10957-012-0027-4
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DOI: https://doi.org/10.1007/s10957-012-0027-4