# 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 elements in a vector space that is still compatible with vector space operations.
• VectorLess is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
• 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
• 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 xiyi 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} such that {"NormCone", n} such that Norm[{x1,…,xn-1}]
• 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 all

## Basic Examples(2)

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:

## 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 xiyi:

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:

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.

#### BibTeX

@misc{reference.wolfram_2020_vectorless, author="Wolfram Research", title="{VectorLess}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/VectorLess.html}", note=[Accessed: 16-April-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2020_vectorless, organization={Wolfram Research}, title={VectorLess}, year={2019}, url={https://reference.wolfram.com/language/ref/VectorLess.html}, note=[Accessed: 16-April-2021 ]}

#### 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