VectorLessEqual

xy or VectorLessEqual[{x,y}]

yields True for vectors of length n if xiyi for all components .

xκy or VectorLessEqual[{x,y},κ]

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

Details

  • VectorLessEqual gives a partial ordering of elements in a vector space that is still compatible with vector space operations.
  • VectorLessEqual is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
  • By using the character , entered as v<= or \[VectorLessEqual], with subscripts vector inequalities can be entered as follows:
  • xyVectorLessEqual[{x,y}]the standard vector inequality
    x_kappayVectorLessEqual[{x,y},κ]vector inequality defined by a cone κ
  • xy remains unevaluated if x or y has non-numeric elements; typically gives True or False otherwise.
  • When x and y are n-vectors, xy yields False if xi>yi for any component .
  • When x is an n-vector and y is a scalar, xy yields True if xiy for all components .
  • 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, VectorLessEqual internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable $MaxExtraPrecision.

Examples

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Basic Examples  (2)

xy yields True when xiyi is True for all i=1,,n:

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xy yields False when xi>yi is False for any i=1,,n:

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Represent a vector inequality:

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

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Scope  (7)

Applications  (8)

Properties & Relations  (3)

Possible Issues  (1)

Introduced in 2019
(12.0)