Positive and Z-operators on closed convex cones

Bishop-Phelps type Theorem for Normed Cones

In this paper the notion of  support points of convex sets  in  normed cones is introduced and it is shown that in a  continuous normed cone, under the appropriate conditions, the set of support points of a  bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones.

Regularities and their relations to error bounds

In this paper, we mainly study various notions of regularity for a finite collection {C1, · · · , Cm} of closed convex subsets of a Banach space X and their relations with other fundamental concepts. We show that a proper lower semicontinuous function f on X has a Lipschitz error bound (resp., Υ-error bound) if and only if the pair {epi(f), X×{0}} of sets in the product space X × R is linearly ...

Lyapunov rank of polyhedral positive operators

IfK is a closed convex cone and if L is a linear operator having L (K) ⊆ K, then L is a positive operator on K and L preserves inequality with respect to K. The set of all positive operators on K is denoted by π (K). If K∗ is the dual of K, then its complementarity set is C (K) := {(x, s) ∈ K ×K | 〈x, s〉 = 0} . Such a set arises as optimality conditions in convex optimization, and a linear oper...

Some Results on Boundedness in Locally Convex Cones

On Polar Cones and Differentiability in Reflexive Banach Spaces

Let $X$ be a  Banach  space, $Csubset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...