Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces
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Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces Nguyen Ngoc Luan1
Received: 15 December 2016 / Accepted: 26 September 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017
Abstract Piecewise linear vector optimization problems in the locally convex Hausdorff topological vector space setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed generalized polyhedral convex sets. If, in addition, the problem is convex, then the efficient solution set and the weakly efficient solution set are the unions of finitely many generalized polyhedral convex sets and they are connected by line segments. Our results develop the preceding ones of Zheng and Yang (Sci. China Ser. A. 51, 1243–1256 2008), and Yang and Yen (J. Optim. Theory Appl. 147, 113–124 2010), which were established in the normed space setting. Keywords Locally convex Hausdorff topological vector space · Generalized polyhedral convex set · Piecewise linear vector optimization problem · Semi-closed generalized polyhedral convex set · Connectedness by line segments Mathematics Subject Classification (2010) 90C29 · 90C30 · 90C48
1 Introduction The intersection of a finite number of closed half-spaces of a finite-dimensional Euclidean space is called a polyhedral convex set (a convex polyhedron in brief). Following Bonnans and Shapiro [2, Definition 2.195], we call a subset of a locally convex Hausdorff topological vector space (lcHtvs) a generalized polyhedral convex set (or a generalized convex polyhedron) if it is the intersection of finitely many closed half-spaces and a closed affine subspace of that topological vector space. If the affine subspace can be chosen as the whole space, the generalized polyhedral convex set (gpcs) is said to be a polyhedral convex set (or a convex polyhedron).
Nguyen Ngoc Luan
[email protected] 1
Department of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam
N. N. Luan
From now on, if not otherwise stated, X and Y are locally convex Hausdorff topological vector spaces. Similarly as in [26], we say that a mapping f : X → Y is a piecewise linear function (or a piecewise affine function) if there exist polyhedral convex sets P1 , . . . , Pm in X, continuous linear mappings T1 , . . . , Tm from X to Y , and vectors b1 , . . . , bm in Y such m that X = Pk and f (x) = Tk x + bk for all x ∈ Pk , k = 1, . . . , m. k=1
We call a subset K ⊂ Y a cone if tK ⊂ K for all t > 0. Given a piecewise linear function f : X → Y , a generalized polyhedral convex set D ⊂ X, and a polyhedral convex cone K ⊂ Y with K = Y , we consider the piecewise linear vector optimization problem MinK {f (x) | x ∈ D} . In the terminology of [7, p. 341], one says that f is a
K-function1
(VP) on D if
(1 − λ)f (x1 ) + λf (x2 ) − f ((1 − λ)x1 + λx2 ) ∈ K for any x1 , x2 in D and λ ∈ [0, 1]. It is clear that
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