Duality Theory on Vector Spaces

TL;DR

The paper extends Fenchel–Rockafellar and Lagrangian duality to vector spaces lacking topological structure, using algebraic interior (core) qualifications and a geometric, coderivative-based approach. It revisits generalized differential calculus in vector spaces, proves a subdifferential maximum rule, and develops conjugate calculus and duality rules (including the Fenchel duality and chain rules) for extended-valued functions. It provides necessary and sufficient optimality conditions for convex minimization and Slater-type results via the algebraic core, and establishes Lagrangian strong duality under core-solid Slater conditions. These results broaden duality theory to purely algebraic settings with potential applications in variational analysis without topology.

Abstract

In this paper, we study the Fenchel-Rockafellar duality and the Lagrange duality in the general frame work of vector spaces without topological structures. We utilize the geometric approach, inspired from its successful application by B. S. Mordukhovich and his coauthors in variational and convex analysis. After revisiting coderivative calculus rules and providing the subdifferential maximum rule in vector spaces, we establish conjugate calculus rules under qualifying conditions through the algebraic interior of the function's domains. Then we develop sufficient conditions which guarantee the Fenchel-Rockafellar strong duality. Finally, after deriving some necessary and sufficient conditions for optimal solutions to convex minimization problems, under a Slater condition via the algebraic interior, we then obtain a sufficient condition for the Lagrange strong duality.

Theorems & Definitions (21)

• Theorem 3.1
• Proposition 3.2
• Theorem 3.3
• Theorem 3.4
• Lemma 3.5
• Theorem 3.6
• Theorem 4.1
• Lemma 4.2
• Theorem 4.3
• Theorem 4.4