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VectorLess

See Also
- Less
- VectorLessEqual
- VectorGreater
- PositiveReals
- Characters
- \[VectorLess]
Related Guides
- Convex Optimization
- Symbolic Vectors, Matrices and Arrays
- See Also
  - Less
  - VectorLessEqual
  - VectorGreater
  - PositiveReals
  - Characters
  - \[VectorLess]
- Related Guides
  - Convex Optimization
  - Symbolic Vectors, Matrices and Arrays

VectorLess

xy or VectorLess[{x,y}]

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

x_κy 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, xy 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, xy is equivalent to . That is, each part of x is less than 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 scalar, xy yields True if x_i<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,…,x_n-1}]<x_n
	"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

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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 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 cone is also given by :

The cuboid is also given by :

Scope (7)

Determine if all of the elements in a vector are strictly positive:

Determine if all components are strictly less than 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 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 :

xy is a (strict) partial order, i.e. irreflexive, asymmetric and transitive:

Irreflexive, i.e. for all elements so no element is related to itself:

Asymmetric, i.e. if then :

Transitive, i.e. if and then :

x_κy 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 :

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