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Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis

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Abstract

This paper is concerned with a unified duality theory for a constrained extremum problem. Following along with the image space analysis, a unified duality scheme for a constrained extremum problem is proposed by virtue of the class of regular weak separation functions in the image space. Some equivalent characterizations of the zero duality property are obtained under an appropriate assumption. Moreover, some necessary and sufficient conditions for the zero duality property are also established in terms of the perturbation function. In the accompanying paper, the Lagrange-type duality, Wolfe duality and Mond–Weir duality will be discussed as special duality schemes in a unified interpretation. Simultaneously, three practical classes of regular weak separation functions will be also considered.

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Acknowledgements

The authors are indebted to Professor F. Giannessi for his helpful advice and valuable discussions, especially for providing references [3, 9, 13] and suggestions on the form of the generalized Lagrange function (5). They would also like to express their gratitude to two anonymous referees for their valuable comments and suggestions on the proof of Lemma 3.1. This research was supported by the National Natural Science Foundation of China (Grant: 11171362) and the Basic and Advanced Research Project of CQCSTC (Grant: cstc2013jcyjA00003).

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

    S. K. Zhu & S. J. Li

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Zhu, S.K., Li, S.J. Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis. J Optim Theory Appl 161, 738–762 (2014). https://doi.org/10.1007/s10957-013-0468-4

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