xy or VectorGreater[{x,y}]

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

xκy or VectorGreater[{x,y},κ]

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


  • VectorGreater gives a partial ordering of elements in a vector space that is still compatible with vector space operations.
  • VectorGreater is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
  • By using the character , entered as v< or \[VectorGreater], with subscripts vector inequalities can be entered as follows:
  • xyVectorGreater[{x,y}]the standard vector inequality
    x_kappayVectorGreater[{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 xi>y 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
    {"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, VectorGreater internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable $MaxExtraPrecision.


open allclose all

Basic Examples  (2)

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

xy 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:

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, !xiyi 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 of where for non-negative x,y with α between 0 and 1:

Applications  (1)

VectorGreater is a fast way to compare many elements:

Introduced in 2019