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A Unified Approach for Constrained Extremum Problems: Image Space Analysis

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Abstract

In this paper, by exploiting the image space analysis we investigate a class of constrained extremum problems, the constraining function of which is set-valued. We show that a (regular) linear separation in the image space is equivalent to the existence of saddle points of Lagrangian and generalized Lagrangian functions and we also give Lagrangian type optimality conditions for the class of constrained extremum problems under suitable generalized convexity and compactness assumptions. Moreover, we consider an exact penalty problem for the class of constrained extremum problems and prove that it is equivalent to the existence of a regular linear separation under suitable generalized convexity and compactness assumptions.

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Acknowledgements

The research was supported by the Natural Science Foundation of China, Project 60804065; the Key Project of Chinese Ministry of Education, Project 211163; and Sichuan Youth Science and Technology Foundation, Project 2012JQ0032.

The authors wish to thank Professor J. Zafarani and two anonymous referees for the careful reading of the paper and valuable comments and suggestions.

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Authors and Affiliations

  1. College of Mathematics and Information, China West Normal University, Nanchong, Sichuan, 637009, China

    J. Li, S. Q. Feng & Z. Zhang

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  1. J. Li
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  2. S. Q. Feng
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  3. Z. Zhang
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Correspondence to J. Li.

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Communicated by Jafar Zafarani.

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Li, J., Feng, S.Q. & Zhang, Z. A Unified Approach for Constrained Extremum Problems: Image Space Analysis. J Optim Theory Appl 159, 69–92 (2013). https://doi.org/10.1007/s10957-013-0276-x

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