A Unified Approach to Extended Real-Valued Functions

TL;DR

This paper introduces a unified framework for extended real-valued functions applicable to various optimization problems, preserving key properties and embedding convex analysis results, thereby advancing the theoretical foundation of optimization theory.

Contribution

It proposes a comprehensive approach to extended real-valued functions that unifies their treatment across different problem types and maintains important analytical properties.

Findings

Preserves continuity and Chebyshev norm in extensions.

Characterizes semicontinuity, convexity, and linearity of extended functions.

Applicable to various image spaces and problem types.

Abstract

Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and into which results of convex analysis can be embedded. Our approach preserves continuity and the Chebyshev norm when extending a functional to the entire space. The basic idea also works for other image spaces. Moreover, we illustrate that extended real-valued functions have to be handled in another way than real-valued functions and characterize semicontinuity, convexity, linearity and related properties of such functions.