Order cone

An order cone or positive cone is a special cone in an ordered vector space . It is defined via the order relation in this vector space. Conversely, however, cones can also be explained as cones of order under certain circumstances and thus define an order relation. Thus, the order cone and the order relation are equivalent in some respects. Every property of the cone then corresponds to an analogous property of the order relation and vice versa.

definition

An ordered vector space is given . Then the amount is called ${\displaystyle (V,\preceq )}$

${\displaystyle K:=\{x\in V\,|\,0\preceq x\}}$

the ordinal cone or the positive cone . It contains all elements that are "positive" with regard to the order relation. Conversely, if a convex cone is in , then becomes through ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V}$

${\displaystyle x\preceq y\iff y-x\in K}$

defines an order relation on which makes an ordered vector space. In this case, too, one calls the order cone. ${\displaystyle V}$ ${\displaystyle (V,\preceq )}$ ${\displaystyle K}$

example

Finite dimensional

On the vector space of the real symmetric matrices is given by ${\displaystyle S^{n}}$ ${\displaystyle n\times n}$

${\displaystyle A\geq _{L}B\iff A-B{\text{ ist positiv semidefinit }}}$

defines the so-called Loewner partial order . The corresponding positive cone is then

${\displaystyle K=\{A\in S^{n}\,|\,A\geq _{L}0\}=\{A\in S^{n}\,|\,A{\text{ ist positiv semidefinit }}\}.}$

Conversely, the Loewner partial order can also be defined using this order cone.

Infinitely dimensional

In the functional space of the continuous in the interval between 0 and 1 functions to define the positive cone ${\displaystyle C[0,1]}$

${\displaystyle K:=\{f\in C[0,1]\,|\,f(t)\geq 0{\text{ für alle }}t\in [0,1]\}}$.

He defines the order

${\displaystyle f\preceq g\iff f(t)\leq g(t){\text{ für alle }}t\in [0,1]}$

and thus makes it an ordered vector space. ${\displaystyle C[0,1]}$

properties

• Every ordering cone defined by an ordered vector space is a 0 cone . This follows directly from the reflexivity of . ${\displaystyle \preceq }$
• Every ordering cone defined by an ordered vector space is a convex cone. This follows from the closeness of the order relation with regard to addition and scalar multiplication. Therefore only convex cones define ordered vector spaces: Weaker cone definitions lead to the loss of these properties.
• The order relation is then antisymmetric, i. H. from and follows , if the order cone pointed , d. H. if . The order relation is then called a strict order. ${\displaystyle a\succeq b}$ ${\displaystyle b\preceq a}$ ${\displaystyle a=b}$ ${\displaystyle K\cap -K=0}$
• The cone , which is dual to the cone of order, defines the so-called dual order on the dual space of . ${\displaystyle V}$

Applications

Order cones and the order relations defined by them are used in the optimization to define generalizations of inequality restrictions. In particular, order cones are somewhat more general than generalized inequalities , since they only assume a convex cone, not a true cone .

The Loewner order mentioned above can be generalized to any C * algebras . If the real vector space is the self-adjoint elements of a C * -algebra , then it is an order cone, which turns into an ordered vector space. The elements of the order cone of the dual order lead to the so-called GNS construction . ${\displaystyle A_{sa}}$ ${\displaystyle A}$ ${\displaystyle \{a^{*}a\,|\,a\in A\}}$ ${\displaystyle A_{sa}}$

literature

• Johannes Jahn: Introduction to the Theory of Nonlinear Optimization. 3. Edition. Springer, Berlin 2007, ISBN 978-3-540-49378-5 .