Functional inequalities and theorems of the alternative involving composite functions

N. Dinh (), G. Vallet () and M. Volle ()

Journal of Global Optimization, 2014, vol. 59, issue 4, 837-863

Abstract: We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form $$ f + g + k\circ H \ge h$$ f + g + k ∘ H ≥ h . The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function $$ f + g + k\circ H$$ f + g + k ∘ 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 The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations.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. Copyright Springer Science+Business Media New York 2014

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; 39B62; 49J52; 46N10 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10898-013-0100-z

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