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VectorLessEqual—Wolfram Language Documentation

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• See Also
  • LessEqual
  • VectorLess
  • VectorGreaterEqual
  • NonNegativeReals
  • Characters
  • \[VectorLessEqual]
• Related Guides
  • Convex Optimization
  • Symbolic Vectors, Matrices and Arrays

xy or VectorLessEqual[{x,y}]

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

x_κy or VectorLessEqual[{x,y},κ]

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

• 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:
• 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:
• Possible cone specifications κ in for matrices x include:
• Possible cone specifications κ in for arrays x include:
• 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.

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 cone is also given by :

The cuboid is also given by :

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:

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

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 :

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. :

Formulate the same problem using matrix variables:

Find an instance of such a matrix:

Check the result:

VectorLessEqual is compatible with a vector space operation:

Adding vectors to both sides for any vector :

Multiplying by positive constants for any :

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

Reflexive, i.e. for all elements :

Antisymmetric, i.e. if and 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 :

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:

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