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Characterizations of the approximate solution sets of nonsmooth optimization problems and its applications

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Abstract

The concept of \(\varphi \)-strongly preinvex functions is introduced, and some properties of \(\varphi \)-strongly preinvex functions are given. Several new and simple characterizations of the approximate solution sets for nonsmooth optimization problems with \(\varphi \)-strong preinvexity are obtained. We establish the relationships between the solutions of Minty-type variational-like inequalities and the approximate solutions of optimization problems. And applying the obtained results, we give the characterizations of the solution sets for the Minty-type variational-like inequalities.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11201379, 11271391, 11201511) and the Fundamental Research Funds for the Central Universities (JBK140926, JBK120504, JBK130401). The authors express their deep gratitude to the anonymous referees for their valuable comments and suggestions which have improved the presentation of the paper.

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Authors and Affiliations

  1. College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

    Caiping Liu

  2. Department of Mathematics, Chongqing Normal University, Chongqing, 400047, China

    Xinmin Yang

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  1. Caiping Liu
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  2. Xinmin Yang
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Correspondence to Caiping Liu.

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Liu, C., Yang, X. Characterizations of the approximate solution sets of nonsmooth optimization problems and its applications. Optim Lett 9, 755–768 (2015). https://doi.org/10.1007/s11590-014-0780-4

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  • DOI: https://doi.org/10.1007/s11590-014-0780-4

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