A maximum theorem for generalized convex functions

Zsolt Pales

doi:10.24193/subbmath.2022.1.02

Authors

Zsolt Pales University of Debrecen

DOI:

https://doi.org/10.24193/subbmath.2022.1.02

Keywords:

maximum theorem, generalized convex function

Abstract

Motivated by the Maximum Theorem for convex functions (in the setting of linear spaces) and for subadditive functions (in the setting of Abelian semigroups), we establish a Maximum Theorem for the class of generalized convex functions, i.e., for functions $f:X\to X$ that satisfy the inequality $f(x\circ y)\leq pf(x)+qf(y)$, where $\circ$ is a binary operation on $X$ and $p,q$ are positive constants. As an application, we also obtain an extension of the Karush--Kuhn--Tucker theorem for this class of functions.

Author Biography

Zsolt Pales, University of Debrecen

Institute of Mathematics, Full Professor

References

begin{thebibliography}{10}

bibitem{Bar02}

A.~Barvinok, emph{{A Course in Convexity}}, {Graduate Studies in Mathematics}, vol.~54, American Mathematical Society, Providence, RI, 2002. MR{1940576}

bibitem{BorLew06}

J.~M. Borwein and A.~S. Lewis, emph{{Convex Analysis and Nonlinear Optimization}}, second ed., {CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC}, vol.~3, Springer, New York, 2006, Theory and examples. MR{2184742}

bibitem{BorVan10}

J.~M. Borwein and J.~D. Vanderwerff, emph{{Convex Functions: Constructions, Characterizations and Counterexamples}}, {Encyclopedia of Mathematics and its Applications}, vol. 109, Cambridge University Press, Cambridge, 2010. MR{2596822}

bibitem{Bri20}

J.~Brinkhuis, emph{{Convex Analysis for Optimization—A Unified Approach}}, {Graduate Texts in Operations Research}, Springer, Cham, 2020. MR{4240269}

bibitem{BriTik05}

J.~Brinkhuis and V.~Tikhomirov, emph{{Optimization: Insights and Applications}}, {Princeton Series in Applied Mathematics}, Princeton University Press, Princeton, NJ, 2005. MR{2168305}

bibitem{Cla13}

F.~Clarke, emph{{Functional Analysis, Calculus of Variations and Optimal Control}}, {Graduate Texts in Mathematics}, vol. 264, Springer, London, 2013. MR{3026831}

bibitem{FucLus81}

B.~Fuchssteiner and W.~Lusky, emph{{Convex Cones}}, {Notas de Matemática [Mathematical Notes]}, vol.~82, North-Holland Publishing Co., Amsterdam-New York, 1981. MR{640719}

bibitem{IofTih79}

A.~D. Ioffe and V.~M. Tihomirov, emph{{Theory of Extremal Problems}}, {Studies in Mathematics and its Applications}, vol.~6, North-Holland Publishing Co., Amsterdam-New York, 1979. MR{528295}

bibitem{MagTik03}

G.~G. {Magaril-Il'yaev} and V.~M. Tikhomirov, emph{{Convex Analysis: Theory and Applications}}, {Translations of Mathematical Monographs}, vol. 222, American Mathematical Society, Providence, RI, 2003. MR{2013877}

bibitem{NicPer18}

C.~P. Niculescu and Lars-Erik Persson, emph{{Convex functions and their applications}}, {CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC}, Springer, Cham, 2018, A contemporary approach, Second edition. MR{3821518}

bibitem{Pop44}

T.~Popoviciu, emph{{Les Fonctions Convexes}}, {Actualités Scientifiques et Industrielles, No. 992}, Hermann et Cie, Paris, 1944. MR{0018705}

bibitem{RobVar73}

A.~W. Roberts and D.~E. Varberg, emph{{Convex Functions}}, {Pure and Applied Mathematics, Vol. 57}, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR{0442824}

bibitem{Zal02}

C.~Zălinescu, emph{{Convex Analysis in General Vector Spaces}}, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR{1921556}

end{thebibliography}