VectorLessEqual

yields True for vectors of length n if x_i≤y_i for all components .

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, xy 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, xy 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.
xy 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, xy yields True if x_i≤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,…,x_n-1}]≤x_n
	"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 (3)

xy yields True when x_i≤y_i is True for all i=1,…,n:

xy yields False when x_i>y_i 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:

The cone is also given by :

The cuboid is also given by :

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:

!xy does not imply xy:

For each component, !x_i≤y_i does imply x_i>y_i:

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:

Optimization over Vector Inequalities (1)

Solving Vector Inequalities (1)

The inequality represents the cuboid Cuboid[p_min,p_max]:

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 ℐ_n=IdentityMatrix[n] and κ="SemidefiniteCone". This finds the set of symmetric matrices with eigenvalues between 1 and 2, i.e. :