VectorLess

xy or VectorLess[{x,y}]

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

xκy or VectorLess[{x,y},κ]

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

Details

  • VectorLess gives a partial ordering of vectors, matrices and arrays that is compatible with vector space operations, so that and imply for all .
  • VectorLess 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 less 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 less than the corresponding part of y for the relation to be true.
  • xy 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 scalar, xy yields True if xi<y for all components .
  • By using the character , entered as v< or \[VectorLess], with subscripts vector inequalities can be entered as follows:
  • xyVectorLess[{x,y}]the standard vector inequality
    x_(kappa)yVectorLess[{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
    {"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 definite matrices
  • Possible cone specifications κ in for arrays x include:
  • "NonNegativeCone"TemplateBox[{}, NonNegativeConeString] such that
  • For exact numeric quantities, VectorLess 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)

x<y yields True when xi < yi is True for all i=1,…,n:

x<y yields False when xi ≥ yi 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 strictly positive:

Determine if all components are strictly less than 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 the boundary where for non-negative x,y with α between 0 and 1:

Applications  (1)

VectorLess is a fast way to compare many elements:

Properties & Relations  (3)

VectorLess is compatible with a vector space operation:

Adding vectors to both sides of for any vector :

Multiplying by positive constants for any :

xy is a (strict) partial order, 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 :

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 comparable elements to :

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_vectorless, author="Wolfram Research", title="{VectorLess}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/VectorLess.html}", note=[Accessed: 11-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_vectorless, organization={Wolfram Research}, title={VectorLess}, year={2019}, url={https://reference.wolfram.com/language/ref/VectorLess.html}, note=[Accessed: 11-February-2025 ]}