# VectorLessEqual xy or VectorLessEqual[{x,y}]

yields True for vectors of length n if xiyi for all components .

xκy or VectorLessEqual[{x,y},κ]

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

# Details  • VectorLessEqual gives a partial ordering of elements in a vector space that is still compatible with vector space operations.
• VectorLessEqual is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.
• 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
• 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 xiy for all components .
• 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 all

## Basic Examples(2)

xy yields True when xiyi 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, !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 where for non-negative x,y with α between 0 and 1:

## Applications(8)

### Basic Applications(1)

VectorLessEqual is a fast way to compare many elements:

### Solving Vector Inequalities(1)

The inequality represents the cuboid Cuboid[pmin,pmax]:

### Integration over Vector Inequality Regions(2)

Integrate over the non-negative quadrant :

Using vector variables:

Integrate over the non-negative orthant:

Integrate over the rectangle :

Using vector variables:

Integrate over the cuboid :

### Matrix Inequalities(3)

Use the standard vector order to represent the set of non-negative matrices:

Give the set of interval bounded matrices:

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

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

Formulate the same problem using matrix variables:

Find an instance of such a matrix:

Check the result:

## Properties & Relations(3)

Vector orders are compatible with with a vector space operation:

Adding vectors to both sides for any vector :

Multiplying by positive constants for any :

Reflexive, i.e. for all elements :

Antisymmetric, i.e. if and then :

Transitive, i.e. if and then :

Vector orders 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 :

## Possible Issues(1)

Vector orders are partial orders, so the negation of is not equivalent to :

Here both and are false:

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

Introduced in 2019
(12.0)