The Fenchel duality formula, The Ekeland variational principle, and Rockafellar's Theorem for maximal monotonicity of subdifferentials. (3rd October 2022)

• Record Type: Journal Article
• Authors: Takahashi, Wataru; Takeuchi, Yukio
• Is Part Of: Optimization. Volume 71:Number 10(2022)
• Page Start: 2769
• Page End: 2787
• Publication Date: 2022-10-03
• Subjects: Rockafellar's theorem for maximal monotonicity of subdifferentials -- the Fenchel duality formula -- the Ekeland variational principle -- Takahashi's nonconvex minimization thorem -- minimization problems; 47H14 -- 47H07 -- 47H11; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2021.1885033
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : In this paper, we first give two versions of the Fenchel duality formula and Brøndsted-Rockafellar's theorem. The versions are useful and the proofs are not so difficult. Next, we present a theorem in a Banach space to reconsider the Ekeland variational principle and Takahashi's nonconvex minimization theorem proved in a complete metric space. Finally, using the version of the Fenchel duality formula, the Ekeland variational principle and the version of Brøndsted-Rockafellar's theorem, we prove a minimization theorem in a Banach space and then present a short proof of well–known Rockafellar's theorem for maximal monotonicity of subdifferentials.