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 elements in a vector space that is still compatible with vector space operations.
  • VectorGreaterEqual is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
  • 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_kappayVectorGreaterEqual[{x,y},κ]vector inequality defined by a cone κ
  • xy remains unevaluated if x or y has a non-numeric element; 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 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

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Basic Examples  (2)

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:

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:

Introduced in 2019
 (12.0)