Functional inequalities and theorems of the alternative involving composite functions

N. Dinh (1) , Guy Vallet (2) , M. Volle

1 Department of Mathematics, International University
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form Mathematical expression. The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function Mathematical expression 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 The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations.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. © 2013 Springer Science+Business Media New York.

Domaines: Equations aux dérivées partielles [math.AP]

N. Dinh, Guy Vallet, M. Volle. Functional inequalities and theorems of the alternative involving composite functions. Journal of Global Optimization, 2013, pp.1-27. ⟨10.1007/s10898-013-0100-z⟩. ⟨hal-00866946⟩