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:
-
VectorLess[{x,y}] the standard vector inequality VectorLess[{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} such that {"NormCone", n} such that Norm[{x1,…,xn-1}]<xn "ExponentialCone" such that "DualExponentialCone" such that {"PowerCone",α} such that {"DualPowerCone",α} such that - Possible cone specifications κ in for matrices x include:
-
"NonNegativeCone" such that {"SemidefiniteCone", n} symmetric positive definite matrices - Possible cone specifications κ in for arrays x include:
-
"NonNegativeCone" 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 allBasic Examples (3)
Scope (7)
Determine if all of the elements in a vector are strictly positive:
Determine if all components are strictly less than 1:
For each component, !xi<yi does imply xi≥yi:
Compare the components of two 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:
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 :
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