Functional inequalities and theorems of the alternative involving composite functions

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$ f + g + k\circ H \ge h$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo>+</mo> <mi>k</mi> <mo>∘</mo> <mi>H</mi> <mo>≥</mo> <mi>h</mi> </mrow> </math> </EquationSource> </InlineEquation>. The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$ f + g + k\circ H$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo>+</mo> <mi>k</mi> <mo>∘</mo> <mi>H</mi> </mrow> </math> </EquationSource> </InlineEquation> 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 The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations.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