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Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions

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Abstract

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.

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Acknowledgements

The authors would like to thank Professor Franco Giannessi for valuable comments and suggestions, which helped to improve the survey paper. This research was partially supported by the Natural Science Foundations of China and China Scholarship Council (Grant Nos. 11571055, 11526165 and 11601437). The fourth author is grateful for the kind hospitality of the institution when part of this work was carried out during a stay in the Department of Mathematics, University of Pisa.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Shengjie Li & Manxue You

  2. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Yangdong Xu

  3. Department of Economics and Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Shengkun Zhu

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  1. Shengjie Li
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  2. Yangdong Xu
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  4. Shengkun Zhu
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Correspondence to Shengjie Li.

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Li, S., Xu, Y., You, M. et al. Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions. J Optim Theory Appl 177, 609–636 (2018). https://doi.org/10.1007/s10957-018-1247-z

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