Applying set optimization to weak efficiency
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Applying set optimization to weak efficiency Giovanni P. Crespi1
· Carola Schrage2
Accepted: 19 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Set-valued extensions of vector-valued functions are used to investigate the relations between weak efficiency and variational inequalities (both Stampacchia and Minty type) which allows to apply the complete lattice framework from set optimization. Since the seminal work of Giannessi, it has been a challenge to generalize scalar results to the vector case. In this effort, some notions of generalized derivatives for vector-valued functions have been introduced, either in the form of set-valued functions or introducing appropriate notions of infinite elements in vector spaces. Switching the focus to set optimization in conlinear spaces, we propose a Dini-type derivative, that keeps the same set-valued form of the optimization problem. Keywords Set optimization · Multiobjective optimization · Variational inequalities · Dini derivative · Weak efficiency
1 Introduction Since the seminal papers by Giannessi Giannessi (1980), Giannessi (1997) one of the issues in (convex) vector optimization has been the use of differentiable variational inequalities to characterize weak efficient solutions of an optimization problem, see e.g. Crespi et al. (2006), Ginchev (2007). The optimization problem is often referred to as primitive (F. Giannessi), since, in these papers, the optimization is studied through the variational inequality, rather than directly. Given a (Fréchet) differentiable, vector-valued, objective function ψ : S ⊆ X → Z , where X and Z are topological vector spaces, and the partial order induced by a closed, convex, pointed cone with nonempty interior C ⊂ Z , the vector optimization problem is min ψ(x), x ∈ S
B
(VOP)
Giovanni P. Crespi [email protected] Carola Schrage [email protected]
1
Dipartimento di Gestione integrata d’impresa, Universitá Carlo Cattaneo - LIUC, Corso Matteotti, 22, 21053 Castellanza, VA, Italy
2
Faculty of Economics and Management, Free University of Bozen-Bolzano, Piazza Università, 1, 39100 Bozen-Bolzano, Italy
123
Annals of Operations Research
A weak efficient solution (VOP) is x0 ∈ S such that ψ(x0 ) is a weakly efficient element in the image set ψ [S] = {ψ(x) | x ∈ S}, i.e., ∀z ∈ ψ [S] :
/ {z} + int C. ψ(x0 ) ∈
Problem (VOP) is referred to as primitive when compared to the Minty variational inequality problem of finding x0 ∈ S s.t. ψ (x), x − x0 ∈ −int C ∀x ∈ S
(MVIP)
or the Stampacchia variational inequality problem of finding x 0 ∈ S s.t. ψ (x0 ), x0 − x ∈ −int C ∀x ∈ S
(SVIP)
The problem has been soon after extended to the nondifferentiable case by using generalized directional derivatives. Relations between the set of weakly efficient solutions of (VOP) and those of the associated (generalized) variational inequalities have been proved in various papers compare e.g. Ansari and Lee (2010), Crespi et al. (2004a), Yang et al. (2004). Crespi et al. (2004b) under mild assumpti
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