Please cite this article in press as: F. García Castaño et al., On geometry of cones and some applications, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa. 2015.06.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.On geometry of cones and some applications AbstractIn this work we prove that in any normed space, the origin is a denting point of a pointed cone if and only if it is a point of continuity for the cone and the closure of the cone in the bidual space respect to the weak * topology is pointed. Other related results and consequences are also stated. For example, a criterion to know whether a cone has a bounded base, an unbounded base, or does not have any base; and a result on the existence of super efficient points in weakly compact sets.

In this paper we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

The aim of this paper is to extend the classical envelope theorem from scalar to vector differential programming. The obtained result allows us to measure the quantitative behaviour of a certain set of optimal values (not necessarily a singleton) characterized to become minimum when the objective function is composed with a positive function, according to changes of any of the parameters which appear in the constraints. We show that the sensitivity of the program depends on a Lagrange multiplier and its sensitivity.

In this paper we provide some equivalences on dentability in normed spaces. Among others we prove: the origin is a denting point of a pointed cone C if and only if it is a point of continuity for such a cone and C * − C * = X * ; x is a denting point of a convex set A if and only if x is a point of continuity and a weakly strongly extreme point of A. We also analize how our results help us to shed some light on several open problems in the literature.Keywords Denting point · point of continuity · dual cone · weakly strongly extreme point Mathematics Subject Classification (2010) MSC 46B40 · MSC 46A40 · MSC 46B20 IntroductionThroughout the paper X will denote a normed space, · the norm of X, X * the dual space of X, 0 X the origin of X, and R + the set of nonnegative real numbers. We call weak the weak topology on X and weak * the weak star topology on its dual X * . A nonempty convex subset C of X is called a cone if αC ⊂ C, ∀α ∈ R + . Fixed a cone C, we define the dual cone for C by C * := {f ∈ X * : f (c) ≥ 0 , ∀c ∈ C} -which is a weak * -closed subset of X * -and the bidual cone for C by C * * := {T ∈ The author M. A.

We show that under a separation property, a $${{{\mathcal {Q}}}}$$ Q -minimal point in a normed space is the minimum of a given sublinear function. This fact provides sufficient conditions, via scalarization, for nine types of proper efficient points; establishing a characterization in the particular case of Benson proper efficient points. We also obtain necessary and sufficient conditions in terms of scalarization for approximate Benson and Henig proper efficient points. The separation property we handle is a variation of another known property and our scalarization results do not require convexity or boundedness assumptions.