Wolfram Language & System Documentation Center

VectorGreater

See Also
- Greater
- VectorGreaterEqual
- VectorLess
- PositiveReals
- Characters
- \[VectorGreater]
Related Guides
- Convex Optimization
- Symbolic Vectors, Matrices and Arrays
- See Also
  - Greater
  - VectorGreaterEqual
  - VectorLess
  - PositiveReals
  - Characters
  - \[VectorGreater]
- Related Guides
  - Convex Optimization
  - Symbolic Vectors, Matrices and Arrays

VectorGreater

xy or VectorGreater[{x,y}]

yields True for vectors of length n if x_i>y_i for all components .

x_κy or VectorGreater[{x,y},κ]

yields True for x and y if , where κ is a proper convex cone.

Details

VectorGreater gives a partial ordering of vectors, matrices and arrays that is compatible with vector space operations, so that and imply for all .
VectorGreater is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
When x and y are -vectors, xy is equivalent to . That is, each part of x is greater than the corresponding part of y for the relation to be true.
When x and y are dimension arrays, xy is equivalent to . That is, each part of x is greater than the corresponding part of y for the relation to be true.
xy remains unevaluated if x or y has non-numeric elements; typically gives True or False otherwise.
When x is an n-vector and y is a numeric scalar, xy yields True if x_i>y for all components .
By using the character , entered as v> or \[VectorGreater], with subscripts vector inequalities can be entered as follows:
VectorGreater[{x,y}] the standard vector inequality

VectorGreater[{x,y},κ] vector inequality defined by a cone κ
In general, one can use a proper convex cone κ to specify a vector inequality. The set is the same as κ.
Possible cone specifications κ in for vectors x include:

	{"NonNegativeCone", n}		such that
	{"NormCone", n}		such that Norm[{x₁,…,x_n-1}]<x_n
	"ExponentialCone"		such that
	"DualExponentialCone"		such that
	{"PowerCone",α}		such that
	{"DualPowerCone",α}		such that

Possible cone specifications κ in for matrices x include:
"NonNegativeCone" such that

{"SemidefiniteCone", n} symmetric positive definite matrices
Possible cone specifications κ in for arrays x include:
"NonNegativeCone" such that
For exact numeric quantities, VectorGreater internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable $MaxExtraPrecision.

Examples

open all close all

Basic Examples (3)

xy yields True when x_i > y_i is True for all i=1,…,n:

xy yields False when x_i ≤ y_i for any i=1,…,n:

Represent a vector inequality:

When v is replaced by numerical vector space elements, the inequality gives True or False:

The cone is also given by :

The cone is also given by :

The cuboid is also given by :

Scope (7)

Determine if all of the elements in a vector are non-negative:

Determine if all components are less than or equal to 1:

!xy does not imply xy:

For each component, !x_i≥y_i does imply x_i<y_i:

Compare the components of two matrices:

Compare symmetric matrices:

Represent the condition that Norm[{x,y}]<=1:

Represent the condition that :

Show the boundary of where for non-negative x,y with α between 0 and 1:

Applications (1)

VectorGreater is a fast way to compare many elements:

Properties & Relations (3)

VectorGreater is compatible with a vector space operation:

Adding vectors to both sides of for any vector :

Multiplying by positive constants for any :

x_κy are (strict) partial orders, i.e. irreflexive, asymmetric and transitive:

Irreflexive, i.e. for all elements so no element is related to itself:

Asymmetric, i.e. if then :

Transitive, i.e. if and then :

x_κy are partial orders but not total orders, so there are incomparable elements:

Neither nor is true, because and are incomparable elements:

The set of vectors and . These are the only comparable elements to :

Top