On Relatively Solid Convex Cones in Real Linear Spaces
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On Relatively Solid Convex Cones in Real Linear Spaces Vicente Novo1
2 · Constantin Zalinescu ˘
Received: 16 April 2020 / Accepted: 17 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Having a convex cone K in an infinite-dimensional real linear space X , Adán and Novo stated (in J Optim Theory Appl 121:515–540, 2004) that the relative algebraic interior of K is nonempty if and only if the relative algebraic interior of the positive dual cone of K is nonempty. In this paper, we show that the direct implication is not true even if K is closed with respect to the finest locally convex topology τc on X , while the reverse implication is not true if K is not τc -closed. However, in the main result of this paper, we prove that the latter implication is true if the algebraic interior of the positive dual cone of K is nonempty; the general case remains an open problem. As a by-product, a result about separation of cones is obtained that improves Theorem 2.2 of the work mentioned above. Keywords Algebraic relative interior · Algebraic dual cone · Algebraic and vectorial closures · Topological dual cone · Convex core topology Mathematics Subject Classification 52A05 · 06F20 · 90C48
1 Preliminary Notions and Remarks In the study of vector optimization problems in locally convex spaces, one needs to consider various properties of the convex cones defining the corresponding ordering; in particular, the solidness of these cones and/or their duals are essential, mainly due to
Communicated by Marc Teboulle.
B
Constantin Z˘alinescu [email protected] Vicente Novo [email protected]
1
Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia, Calle Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
2
Octav Mayer Institute of Mathematics, Ia¸si Branch of Romanian Academy, 700505 Iasi, Romania
123
Journal of Optimization Theory and Applications
some scalarization processes. Of course, the properties of the ordering cone are closely linked to the properties of its dual cones. Studies in this direction became natural in the framework of linear spaces too (see, for example, [1,2] and the references therein). In the sequel, by a linear space we mean a nontrivial real linear space. Having X a linear space and ∅ = A ⊆ X we denote by span A, aff A, conv A the linear hull, affine hull and convex hull of A, respectively; lin0 A is the linear subspace parallel to aff A, that is, lin0 A = span(A − A) (= span(A − a) for all a ∈ A). The algebraic interior (or core) and the relative algebraic interior (or intrinsic core) of A are the sets cor A := {a ∈ A : (∀x ∈ X ) (∃δ > 0) (∀λ ∈ [0, δ]) a + λx ∈ A}, icr A := {a ∈ A : (∀x ∈ lin0 A) (∃δ > 0) (∀λ ∈ [0, δ]) a + λx ∈ A}, respectively; the set A is called (relatively) solid if its (relative) algebraic interior is nonempty. Of course, cor A =
icr A, if aff A = X , ∅, otherwise.
If A is convex, then a ∈ cor A ⇐⇒ [(∀x ∈ X ) (∃δ > 0) a + δx ∈ A], a ∈ icr A ⇐⇒ [(∀x ∈ A) (∃λ > 0) (1 + λ)a − λ
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