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02(39) (2020) 55-57 Overview of Legendre-Fenchel duality Tổng quan về đối ngẫu Legendre-Fenchel Tina Mai a,b,∗ Mai Ti Na a b ∗ a. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam Viện Nghiên cứu và Phát triển Công nghệ cao, Đại học Duy Tân, Đà Nẵng, 550000, Việt Nam b. Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam Khoa Khoa học Tự nhiên, Đại học Duy Tân, Đà Nẵng, 550000, Việt Nam (Ngày nhận bài: 22/10/2019, ngày phản biện xong: 29/10/2019, ngày chấp nhận đăng: 4/5/2020) Abstract We give some overview of Legendre-Fenchel duality. Keywords: Legendre-Fenchel duality. Tóm tắt Chúng tôi đưa ra một vài tổng quan về đối ngẫu Legendre-Fenchel. Từ khóa: Đối ngẫu Legendre-Fenchel. 1. Introduction Legendre-Fenchel duality plays a helpful role in convex optimization. Herein, we introduce some overview of Legendre-Fenchel duality, with an eye toward later applications in nonlinear elasticity. The basic tool here is functional analysis. X ∗ , with the associated duality X ∗ 〈·, ·〉 X . The bidual space of X is denoted by X ∗∗ . In case X is a reflexive Banach space, X ∗∗ will coincide with X by means of the usual canonical isometry. Let A be a subset of X . The indicator function of A is defined by I A (x) := 2. Preliminaries In this paper, we work with real field. The notations here are as introduced in [1]. The dual space of normed vector space X is denoted by * ( 0 if x ∈ A , +∞ if x ∉ A . A function g : X → R ∪ {+∞} is proper if {x ∈ X |g (x) < +∞} , ;. Let Σ be a normed vector space and let g : Σ → R ∪ {+∞} be a proper function. The Corresponding Author: Tina Mai; Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam; Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam. Email: maitina@duytan.edu.vn Tina Mai / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 02(39) (2020) 55-57 56 Legendre-Fenchel transform of g is the function g ∗ : Σ∗ → R ∪ {+∞} defined by g ∗ : ² ∈ Σ∗ → g ∗ (²) := sup{Σ∗ 〈², σ〉Σ − g (σ)} . Theorem 3.2 ([1]). Let Σ and V be two reflexive Banach spaces, and given g : Σ → R ∪ {+∞} and h : V ∗ → R ∪ {+∞} two proper, strictly convex, and lower semi-continuous functions, let Λ : Σ → V ∗ be a linear and continuous mapping. Let the function G : Σ → R ∪ {+∞} be defined by. G : σ ∈ Σ → G(σ) := g (σ) + h(Λσ) . σ∈Σ In nonlinear elasticity, σ and ² represent the traditional stress and strain, respectively. Finally, let the two Lagrangians associated with the minimization problem (P ). L : Σ × Σ∗ → {−∞} ∪ R ∪ {+∞} 3. Legendre-Fenchel duality and We consider a given reflexive Banach space Σ. The next theorem summaries some basic properties of the Legendre-Fenchel transform. We refer the readers to [1, 2] for the statement and proof. Theorem 3.1 ([1, 2]). Let Σ be a reflexive Banach space, and given g : Σ → R∪{+∞} a proper, strictly convex, and lower semi-continuous function. Then, the Legendre-Fenchel transform g ∗ : Σ∗ → R ∪ {+∞} of g is also proper, strictly convex, and lower semi-continuous. Let L̃ : Σ × V → {−∞} ∪ R ∪ {+∞} be defined by L : (σ, ²) ∈ Σ × Σ∗ → L (σ, ²) where L (σ, ²) := Σ∗ 〈², σ〉Σ − g ∗ (²) + h(Λσ) , and L̃ : (σ, v ) ∈ Σ × V → L̃ (σ, v ) where g ∗∗ : σ ∈ Σ∗∗ → g ∗∗ (σ) := sup {Σ∗ 〈², σ〉Σ −g ∗ (²)} L̃ (σ, v ) := g (σ) + V ∗ 〈Λσ, v 〉V − h ∗ (v ) . ²∈Σ∗ denote the Legendre-Fenchel transform of g ∗ . Then, (with X ∗∗ ≡ X ), Then, inf G(σ) = inf sup L (σ, ²) = inf sup L̃ (σ, v ) . σ∈Σ g ∗∗ =g. σ∈Σ ²∈Σ∗ σ∈Σ v ∈V The equality g ∗∗ = g forms the FenchelMoreau theorem. In our case, as in [1], the dual problem corresponding to the first inf-sup problem found in Theorem 3.2 is defined as problem (P ∗ ) with Given a minimization problem (P ) with sup G ∗ (²) , inf G(σ) , σ∈Σ ²∈Σ∗ (1) provided a function G : Σ → R ∪ {+∞} of the specific form given in Theorem 3.2, the following result will be the basis for defining two different dual problems of problem (P ) with (1). The proof is based on Theorem 3.1 and can be found in [1]. where G ∗ (²) := inf {Σ∗ 〈², σ〉Σ +h(Λσ)}−g ∗ (²) σ∈Σ ∀² ∈ Σ∗ . (2) The dual problem corresponding to the second sup-inf problem is defined as problem (P̃ ∗ ) with sup G̃ ∗ (v ) , v ∈V Tina Mai / Tạp chí Khoa học và Công nghệ Đại học Duy Tân 02(39) (2020) 55-57 57 where optimization. We wish to later apply this knowlG̃ ∗ (v ) := inf {g (σ)+Σ∗ 〈Λσ, v 〉V }−h ∗ (v ) ∀v ∈ V . edge to nonlinear elasticity in three-dimensional σ∈Σ settings. The main tool here is functional analy(3) sis. A key matter then includes deciding whether the infimum found in problem (P ) with (1) is equal to the supremum found in either one of its References dual problems. If this is the case, the next issue consists of identifying whether the Lagrangian L has a [1] P. G. Ciarlet, G. Geymonat, and F. Krasucki. A new saddle-point (T , E ) ∈ Σ × Σ∗ . duality approach to elasticity. Mathematical Mod4. Conclusions In this paper, we introduce some overview of Legendre-Fenchel duality, in the spirit of convex els and Methods in Applied Sciences, 22(1):21 pages, 2012. DOI: 10.1142/S0218202512005861. [2] I. Ekeland and R. Temam. Convex analysis and variational problems. Philadelphia : SIAM, Society for Industrial and Applied Mathematics, 1999.