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Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions

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Abstract

The paper is devoted to the study of a new class of conic constrained optimization problems with objectives given as differences of a composite function and a convex function. We first introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we provide necessary and sufficient conditions for several versions of Farkas lemmas to hold. Similarly, we provide characterizations for conic constrained optimization problems to have the strong or stable strong dualities such as Lagrange, Fenchel–Lagrange or Toland–Fenchel–Lagrange duality.

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Acknowledgements

The authors thank the editor and the referees for their valuable comments and constructive suggestions which improved the presentation of this manuscript. Research work of the first author is supported in part by the National Natural Science Foundation of China (Grant 11461027), Hunan Provincial Natural Science Foundation of China (Grant 2016JJ2099) and the Scientific Research Fund of Hunan Provincial Education Department (Grant 17A172). The second authors is supported in part by the Scientific Research Fund of Hunan Provincial Education Department (Grant 15C1156).

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  1. College of Mathematics and Statistics, Jishou University, Jishou, 416000, People’s Republic of China

    D. H. Fang & Y. Zhang

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  1. D. H. Fang
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  2. Y. Zhang
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Correspondence to D. H. Fang.

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Fang, D.H., Zhang, Y. Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions. J Optim Theory Appl 176, 351–376 (2018). https://doi.org/10.1007/s10957-018-1219-3

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  • DOI: https://doi.org/10.1007/s10957-018-1219-3

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