VectorLessEqual
✖
VectorLessEqual
xy or VectorLessEqual[{x,y}]
yields True for vectors of length n if xi≤yi for all components .
xκy or VectorLessEqual[{x,y},κ]
yields True for x and y if y-x∈κ, where κ is a proper convex cone.
Details


- VectorLessEqual gives a partial ordering of vectors, matrices and arrays that is compatible with vector space operations, so that
and
imply
for all
.
- VectorLessEqual is typically used to specify vector inequalities for constrained optimization, inequality solving and integration. It is also used to define minimal elements in vector optimization.
- When x and y are
-vectors, xy is equivalent to
. That is, each part of x is less than or equal to 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 or equal to 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 numeric scalar, xy yields True if xi≤y for all components
.
- By using the character , entered as
v<=
or \[VectorLessEqual], with subscripts vector inequalities can be entered as follows:
-
VectorLessEqual[{x,y}] the standard vector inequality VectorLessEqual[{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
or
{"PowerCone",α} such that
{"DualPowerCone",α} such that
- Possible cone specifications κ in
for matrices x include:
-
"NonNegativeCone" such that
{"SemidefiniteCone", n} symmetric positive semidefinite matrices - Possible cone specifications κ in
for arrays x include:
-
"NonNegativeCone" 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
open allclose allBasic Examples (3)Summary of the most common use cases
xy yields True when xi≤yi is True for all i=1,…,n:

https://wolfram.com/xid/0dqx2b5n56y-bm4a60

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

https://wolfram.com/xid/0dqx2b5n56y-bx19bo

Represent a vector inequality:

https://wolfram.com/xid/0dqx2b5n56y-j6uv96

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

https://wolfram.com/xid/0dqx2b5n56y-esdvy6


https://wolfram.com/xid/0dqx2b5n56y-g1838z


https://wolfram.com/xid/0dqx2b5n56y-lqink


https://wolfram.com/xid/0dqx2b5n56y-dog27u


https://wolfram.com/xid/0dqx2b5n56y-dlrjpq

Scope (7)Survey of the scope of standard use cases
Determine if all of the elements in a vector are non-negative:

https://wolfram.com/xid/0dqx2b5n56y-egz77b

https://wolfram.com/xid/0dqx2b5n56y-dsvgs7

Determine if all components are less than or equal to 1:

https://wolfram.com/xid/0dqx2b5n56y-f21cxf


https://wolfram.com/xid/0dqx2b5n56y-cwq9cq

https://wolfram.com/xid/0dqx2b5n56y-oso6qe


https://wolfram.com/xid/0dqx2b5n56y-cu0xkj

For each component, !xi≤yi does imply xi>yi:

https://wolfram.com/xid/0dqx2b5n56y-d7s40k

Compare the components of two matrices:

https://wolfram.com/xid/0dqx2b5n56y-bsdwb5

https://wolfram.com/xid/0dqx2b5n56y-do2gvh


https://wolfram.com/xid/0dqx2b5n56y-g9emso

https://wolfram.com/xid/0dqx2b5n56y-g63km1


https://wolfram.com/xid/0dqx2b5n56y-y63je


https://wolfram.com/xid/0dqx2b5n56y-kgkn7p


https://wolfram.com/xid/0dqx2b5n56y-j308dt

Represent the condition that Norm[{x,y}]<=1:

https://wolfram.com/xid/0dqx2b5n56y-vvt94


https://wolfram.com/xid/0dqx2b5n56y-b9zhpw

Represent the condition that :

https://wolfram.com/xid/0dqx2b5n56y-bvbxar


https://wolfram.com/xid/0dqx2b5n56y-mhpo9n

Show where for non-negative x,y with α between 0 and 1:

https://wolfram.com/xid/0dqx2b5n56y-h8dfui

Applications (8)Sample problems that can be solved with this function
Basic Applications (1)
VectorLessEqual is a fast way to compare many elements:

https://wolfram.com/xid/0dqx2b5n56y-bb90pl

https://wolfram.com/xid/0dqx2b5n56y-i31j70


https://wolfram.com/xid/0dqx2b5n56y-fbt5y9

Optimization over Vector Inequalities (1)

https://wolfram.com/xid/0dqx2b5n56y-g1owcx

Solving Vector Inequalities (1)
The inequality represents the cuboid Cuboid[pmin,pmax]:

https://wolfram.com/xid/0dqx2b5n56y-iaxemd

https://wolfram.com/xid/0dqx2b5n56y-dwvl2b


https://wolfram.com/xid/0dqx2b5n56y-gleh2

Integration over Vector Inequality Regions (2)
Integrate over the non-negative quadrant
:

https://wolfram.com/xid/0dqx2b5n56y-cb1if4


https://wolfram.com/xid/0dqx2b5n56y-mzkeix


https://wolfram.com/xid/0dqx2b5n56y-j2b2zp

Integrate over the non-negative orthant:

https://wolfram.com/xid/0dqx2b5n56y-bia6yl

Integrate over the rectangle
:

https://wolfram.com/xid/0dqx2b5n56y-cqziac


https://wolfram.com/xid/0dqx2b5n56y-iwp28j


https://wolfram.com/xid/0dqx2b5n56y-eae2bv

https://wolfram.com/xid/0dqx2b5n56y-davu6j

Matrix Inequalities (3)
Use the standard vector order to represent the set of non-negative matrices:

https://wolfram.com/xid/0dqx2b5n56y-ni6qzt

Give the set of interval bounded matrices:

https://wolfram.com/xid/0dqx2b5n56y-c5aac5

Use the semidefinite cone to define the set of symmetric positive semidefinite matrices:

https://wolfram.com/xid/0dqx2b5n56y-bjp6o5

Define the set of symmetric matrices with smallest eigenvalue and largest eigenvalue
by using
, where ℐn=IdentityMatrix[n] and κ="SemidefiniteCone". This finds the set of symmetric matrices
with eigenvalues between 1 and 2, i.e.
:

https://wolfram.com/xid/0dqx2b5n56y-yiv5f

https://wolfram.com/xid/0dqx2b5n56y-bozlz4

Formulate the same problem using matrix variables:

https://wolfram.com/xid/0dqx2b5n56y-be5s7h

Find an instance of such a matrix:

https://wolfram.com/xid/0dqx2b5n56y-kl59c


https://wolfram.com/xid/0dqx2b5n56y-etiws0

Properties & Relations (3)Properties of the function, and connections to other functions
VectorLessEqual is compatible with a vector space operation:

https://wolfram.com/xid/0dqx2b5n56y-djby60
Adding vectors to both sides for any vector
:

https://wolfram.com/xid/0dqx2b5n56y-cbatcv

https://wolfram.com/xid/0dqx2b5n56y-c76en


https://wolfram.com/xid/0dqx2b5n56y-fidirk

Multiplying by positive constants for any
:

https://wolfram.com/xid/0dqx2b5n56y-8csn5

https://wolfram.com/xid/0dqx2b5n56y-gx7g3f


https://wolfram.com/xid/0dqx2b5n56y-coe8bo

xy is a (non-strict) partial order, i.e. reflexive, antisymmetric and transitive:

https://wolfram.com/xid/0dqx2b5n56y-dxz3jr
Reflexive, i.e. for all elements
:

https://wolfram.com/xid/0dqx2b5n56y-mrtvfb


https://wolfram.com/xid/0dqx2b5n56y-dtoi0n

Antisymmetric, i.e. if and
then
:

https://wolfram.com/xid/0dqx2b5n56y-6r4hd

Transitive, i.e. if and
then
:

https://wolfram.com/xid/0dqx2b5n56y-l1sy7g

xκy are partial orders but not total orders, so there are incomparable elements:

https://wolfram.com/xid/0dqx2b5n56y-ixvup8
Neither nor
is true, because
and
are incomparable elements:

https://wolfram.com/xid/0dqx2b5n56y-bttlje

The set of vectors and
. These are the comparable elements to
:

https://wolfram.com/xid/0dqx2b5n56y-nidqc5

Possible Issues (1)Common pitfalls and unexpected behavior
Vector orders are partial orders, so the negation of is not equivalent to
:

https://wolfram.com/xid/0dqx2b5n56y-ejohqm

https://wolfram.com/xid/0dqx2b5n56y-pj8m

Visualize and
. The difference of these sets consists of incomparable elements:

https://wolfram.com/xid/0dqx2b5n56y-m8swzt

Wolfram Research (2019), VectorLessEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorLessEqual.html.
Text
Wolfram Research (2019), VectorLessEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorLessEqual.html.
Wolfram Research (2019), VectorLessEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/VectorLessEqual.html.
CMS
Wolfram Language. 2019. "VectorLessEqual." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorLessEqual.html.
Wolfram Language. 2019. "VectorLessEqual." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VectorLessEqual.html.
APA
Wolfram Language. (2019). VectorLessEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorLessEqual.html
Wolfram Language. (2019). VectorLessEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VectorLessEqual.html
BibTeX
@misc{reference.wolfram_2025_vectorlessequal, author="Wolfram Research", title="{VectorLessEqual}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/VectorLessEqual.html}", note=[Accessed: 29-April-2025
]}
BibLaTeX
@online{reference.wolfram_2025_vectorlessequal, organization={Wolfram Research}, title={VectorLessEqual}, year={2019}, url={https://reference.wolfram.com/language/ref/VectorLessEqual.html}, note=[Accessed: 29-April-2025
]}