A maximum theorem for generalized convex functions

TL;DR

This paper extends the Maximum Theorem to a broad class of generalized convex functions defined on algebraic structures, providing new theoretical insights and an extension of the KKT theorem.

Contribution

It introduces a Maximum Theorem for generalized convex functions satisfying a specific inequality, broadening the scope beyond classical convex and subadditive functions.

Findings

Established a maximum theorem for generalized convex functions.

Extended the Karush--Kuhn--Tucker theorem to this class.

Provided theoretical foundations for optimization in algebraic structures.

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 $f:X\to \mathbb{R}$ 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.