# 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 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, xy 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, xy is equivalent to . That is, each part of x is less than 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 scalar, xy yields True if xi<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,…,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(3)

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:

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:

!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:

## 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 :

xy 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 :

xκy 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 :