Abstract
We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form \( f + g + k\circ H \ge h\). The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function \( f + g + k\circ H\) that are satisfied in various favorable circumstances. The results are then applied to the contexts of alternative type theorems involving composite and DC functions. The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations. As applications of these results, nonconvex optimization problems with composite functions, convex composite problems with conic constraints are examined at the end of the paper. There, strong duality, stable strong duality results for these classes of problems are established. Farkas-type results and stable form of these results for the corresponding systems involving composite functions are derived as well.
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Acknowledgments
The authors would like to thank the two anonymous referees for the valuable comments that help to improve the quality of the manuscript. Parts of the work of the first author was initiated during his visit at the Laboratory of Applied Mathematics, University of Pau, to which he is grateful to the hospitality received. His work is also supported by the National Foundation for Sciences & Technology Development (NAFOSTED, Vietnam).
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Dinh, N., Vallet, G. & Volle, M. Functional inequalities and theorems of the alternative involving composite functions. J Glob Optim 59, 837–863 (2014). https://doi.org/10.1007/s10898-013-0100-z
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DOI: https://doi.org/10.1007/s10898-013-0100-z
Keywords
- Functional inequalities
- Alternative type theorems
- Farkas-type results
- Stable strong duality
- Stable Farkas lemma
- Set containments
- Nonconvex composite optimization problems
- Convex composite problems with conic constraints