A convex, finite and lower semicontinuous function with empty subdifferential

2024/09/27 by Gerd Wachsmuth, Wachsmuth, Gerd
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Equations Stability Results #Optimization and Control (math.OC) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2409.18650

openalex publication_date 2024/09/27 · openalex created_date 2024/10/27 · openalex updated_date 2026/07/28

Abstract

We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal counterexample to various statements in convex analysis in which completeness is required.

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