Optimality and duality in nonsmooth composite vector optimization and applications
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Optimality and duality in nonsmooth composite vector optimization and applications Thai Doan Chuong1,2
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract This article is devoted to the study of a nonsmooth composite vector optimization problem (P for brevity). We apply some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of (P). Sufficient conditions for the existence of such solutions to (P) are also provided by means of proposing the use of (strictly) generalized convex composite vector functions with respect to a cone. We also state a dual problem to (P) and explore weak, strong and converse duality relations. In addition, applications to a multiobjective approximation problem and a composite multiobjective problem with linear operators are deployed. Keywords Necessary/sufficient conditions · Duality · Composite vector optimization · Generalized convexity · Limiting/Mordukhovich subdifferential Mathematics Subject Classification 49K99 · 65K10 · 90C29 · 90C46
1 Introduction Let F : X → W and f : W → Y be vector functions between finite-dimensional spaces. Given a pointed (i.e., K ∩ (−K ) = {0}) closed convex cone K ⊂ Y , we consider a composite vector optimization problem of the form: min K ( f ◦ F)(x) | x ∈ C . (P) Here, the feasible set C is defined by C := x ∈ X | (g ◦ G)(x) ∈ −S ,
(1.1)
Dedicated to Professor Gue Myung Lee on the occasion of his 65th birthday.
B
Thai Doan Chuong [email protected]
1
Optimization and Applications Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
123
Annals of Operations Research
where G : X → V , g : V → Z are vector functions between finite-dimensional spaces and S ⊂ Z is a nonempty closed convex cone. From now on, we always assume that F, G, f , g are locally Lipschitz at the corresponding points under consideration. The problem (P) has a quite general formulation, which provides a unified framework for examining various multiobjective/vector optimization problems (see e.g., Jeyakumar and Yang 1993; Tang and Zhao 2013). For example, when X = W = V and F and G are identical maps, the problem (P) reduces to the following conic vector optimization problem: min K f (x) | x ∈ X , g(x) ∈ −S . (CP) In particular, if X = W = V , F and G are identical maps, K is the nonnegative orthant in p R p (i.e., K := R+ ), S := (α1 , . . . , αm , β1 , . . . , βn ) ∈ Rm+n | αi ≥ 0, i = 1, . . . , m, β j = 0, j = 1, . . . , n (1.2) and f : X → Y := R p and g : X → Z := Rm+n are given respectively by f (x) := ( f 1 (x), . . . , f p (x)), g(x) := (g1 (x), . . . , gm (x), h 1 (x), . . . , h n (x)), x ∈ X , (1.3) then the problem (P) collapses to a (standard) multiobjective/vector optimization problem of the form: (MP) minR p ( f 1 (x), . . . , f p (x)) | x ∈ X , gi (x) ≤ 0, i = 1, . . . , m, + h j (x) = 0, j = 1, . . . , n , where f
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