Generalized Separation of Collections of Sets

TL;DR

The paper develops a unifying generalized separation framework for collections of sets in normed spaces using abstract product norms on $X^{n-1}$ and $X^n$, showing that existing extremal-principle-type results are recovered as special cases of the main theorem by choosing appropriate product norms. Its approach combines the Ekeland variational principle with subdifferential calculus to derive dual necessary conditions expressed via Fréchet/Clarke normal cones on product spaces, with extensions to Asplund spaces. The central contributions include a general separation theorem that encompasses prior results, a multi-parameter norm perspective, and a direct application to dual characterizations of approximate stationarity and transversality properties through product-norm extensions. This framework broadens the scope of generalized separation, providing unified tools for multiplier rules and robust optimality conditions in nonconvex settings. The results hold both in finite and infinite-dimensional settings, offering practical impact for variational analysis and optimization theory.

Abstract

We show that the existing generalized separation statements including the conventional extremal principle and its extensions differ {in the ways norms on product spaces are defined}. We prove a general separation statement with arbitrary product norms covering the existing results of this kind. The proof is divided into a series of claims and exposes the key steps and arguments used when proving generalized separation statements. As an application, we prove dual necessary (sufficient) conditions for an abstract product norm extension of the approximate stationarity (transversality) property.

Theorems & Definitions (28)

• Definition 1.1: Extremality
• Theorem 1.2: Extremal principle
• Theorem 1.3
• Remark 1
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Example 2.4
• Proposition 2.5