Notes about extended real- and set-valued functions

Abstract

An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this, definitions and results for directional derivatives, subdifferentials and Legendre--Fenchel conjugates for extended real-valued functions are given which admit to include the proper as well as the improper case. For set-valued functions, scalar representation theorems and a new conjugation theory are established. The common denominator is that the appropriate image spaces for set-valued functions share fundamental structures with the extended real numbers: They are order complete, residuated monoids with a multiplication by non-negative real numbers.

Cited by in corpus (4)

• Lagrange Duality in Set Optimization
• Scalar Representation and Conjugation of Set-Valued Functions
• A Minty variational principle for set optimization
• Conjugate Duality of Set--Valued Functions