VectorGreaterEqual

xy or VectorGreaterEqual[{x,y}]

yields True for vectors of length n if xi≥yi for all components .

xκy or VectorGreaterEqual[{x,y},κ]

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

Details

  • VectorGreaterEqual gives a partial ordering of vectors, matrices and arrays that is compatible with vector space operations, so that and imply for all .
  • VectorGreaterEqual is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
  • When x and y are -vectors, xy is equivalent to . That is each part of x is greater or equal than the corresponding part of y for the relation to be true.
  • When x and y are dimension arrays, xy is equivalent to . That is each part of x is greater or equal than the corresponding part of y for the relation to be true.
  • xy remains unevaluated if x or y has a non-numeric element; typically gives True or False otherwise.
  • When x is an n-vector and y is a numeric scalar, xy yields True if xi≥y for all components .
  • By using the character , entered as v>= or \[VectorGreaterEqual], with subscripts vector inequalities can be entered as follows:
  • xyVectorGreaterEqual[{x,y}]the standard vector inequality
    x_(kappa)yVectorGreaterEqual[{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}TemplateBox[{n}, NonNegativeConeList] such that
    {"NormCone", n}TemplateBox[{n}, NormConeList] such that Norm[{x1,…,xn-1}]≤xn
    "ExponentialCone"TemplateBox[{}, ExponentialConeString] such that
    "DualExponentialCone"TemplateBox[{}, DualExponentialConeString] such that or
    {"PowerCone",α}TemplateBox[{alpha}, PowerConeList] such that
    {"DualPowerCone",α}TemplateBox[{alpha}, DualPowerConeList] such that
  • Possible cone specifications κ in for matrices x include:
  • "NonNegativeCone"TemplateBox[{}, NonNegativeConeString] such that
    {"SemidefiniteCone", n}TemplateBox[{n}, SemidefiniteConeList]symmetric positive semidefinite matrices
  • Possible cone specifications κ in for arrays x include:
  • "NonNegativeCone"TemplateBox[{}, NonNegativeConeString] such that
  • For exact numeric quantities, VectorGreaterEqual internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable $MaxExtraPrecision.

Examples

open allclose all

Basic Examples  (3)

xy yields True when xi≥yi is True for all i=1,…,n:

xy yields False when xi<yi is False 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:

!xy does not imply xy:

For each component, !xi≥yi does imply xi<yi:

Compare the components of two matrices:

Compare symmetric matrices:

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

Represent the condition that :

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

Applications  (1)

VectorGreaterEqual is a fast way to compare many elements:

Properties & Relations  (3)

VectorGreaterEqual is compatible with a vector space operation:

Adding vectors to both sides of for any vector :

Multiplying by positive constants for any :

xκy are (non-strict) partial orders, i.e. reflexive, antisymmetric and transitive:

Reflexive, i.e. for all elements :

Antisymmetric, i.e. if and then :

Transitive, i.e. if and then :

xκy 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 :

Wolfram Research (2019), VectorGreaterEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorGreaterEqual.html.

Text

Wolfram Research (2019), VectorGreaterEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorGreaterEqual.html.

CMS

Wolfram Language. 2019. "VectorGreaterEqual." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorGreaterEqual.html.

APA

Wolfram Language. (2019). VectorGreaterEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorGreaterEqual.html

BibTeX

@misc{reference.wolfram_2024_vectorgreaterequal, author="Wolfram Research", title="{VectorGreaterEqual}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/VectorGreaterEqual.html}", note=[Accessed: 21-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_vectorgreaterequal, organization={Wolfram Research}, title={VectorGreaterEqual}, year={2019}, url={https://reference.wolfram.com/language/ref/VectorGreaterEqual.html}, note=[Accessed: 21-November-2024 ]}