({{, x\[VectorLess]y or (RefLink[VectorLess, paclet:ref/VectorLess])[{x,y}]
yields (RefLink[True, paclet:ref/True]) for vectors of length n if (Subscript[x, i])<(Subscript[y, i]) for all components 1≤i≤n.}, {, x(Subscript[\[VectorLess], κ])y or (RefLink[VectorLess, paclet:ref/VectorLess])[{x,y},κ]
yields (RefLink[True, paclet:ref/True]) for x and y if y-x∈interior(κ), where κ is a proper convex cone.}})

OpenersNotesSection(Spacer1)Details

(RefLink[VectorLess, paclet:ref/VectorLess]) gives a partial ordering of elements in a vector space that is still compatible with vector space operations.

(RefLink[VectorLess, paclet:ref/VectorLess]) is typically used to specify vector inequalities for constrained optimization, inequality solving and integration.

By using the character \[VectorLess], entered as (Key0[esc]) v<(Key0[esc]) or (RefLink[\[VectorLess], paclet:ref/character/VectorLess]), with subscripts vector inequalities can be entered as follows:

x\[VectorLess]y remains unevaluated if x or y has non-numeric elements, typically gives (RefLink[True, paclet:ref/True]) or (RefLink[False, paclet:ref/False]) otherwise.

When x and y are n-vectors x\[VectorLess]y yields (RefLink[False, paclet:ref/False]) if (Subscript[x, i])>(Subscript[y, i]) for any component 1≤i≤n.

When x is an n-vector and y is a scalar, x\[VectorLess]y yields (RefLink[True, paclet:ref/True]) if (Subscript[x, i])<y for all components 1≤i≤n.

Possible cone specifications κ in 0(Subscript[\[VectorLess], κ])x for vectors x include:

Possible cone specifications κ in 0(Subscript[\[VectorLess], κ])x for matrices x include:

Possible cone specifications κ in 0(Subscript[\[VectorLess], κ])x for arrays x include:

For exact numeric quantities, (RefLink[VectorLess, paclet:ref/VectorLess]) internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable (RefLink[$MaxExtraPrecision, paclet:ref/$MaxExtraPrecision]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (10)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

x<y yields (RefLink[True, paclet:ref/True]) when (Subscript[x, i]) < (Subscript[y, i]) is (RefLink[True, paclet:ref/True]) for all i=1,…,n:

{1,2,3}\[VectorLess]{2,3,4}

x<y yields (RefLink[False, paclet:ref/False]) when (Subscript[x, i]) ≥ (Subscript[y, i]) for any i=1,…,n:

{1,2,3}\[VectorLess]{2,3,3}

Represent a vector inequality:

vi = {1,2,3}\[VectorLess]v

When v is replaced by numerical vector space elements, the inequality gives (RefLink[True, paclet:ref/True]) or (RefLink[False, paclet:ref/False]):

vi /. v->3

vi /. v->{2,4,5}

OpenersExampleSection1(Spacer1)Scope  (7)

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

v = RandomReal[1,10];

0\[VectorLess]v

Determine if all components are strictly less than 1:

v\[VectorLess]1

!x\[VectorLess]y does not imply x\[VectorGreater]y:

x = {1,0};
y = {0,1};

!x\[VectorLess]y

x\[VectorGreater]y

For each component, !(Subscript[x, i])<(Subscript[y, i]) does imply (Subscript[x, i])≥(Subscript[y, i]):

TableForm[{Not /@ Thread[x<y], Thread[x≥y]}, TableHeadings->{{HoldForm[!(Subscript[x, i])<(Subscript[y, i])], HoldForm[(Subscript[x, i])≥(Subscript[y, i])]},{1,2}}]

Compare the components of two matrices:

m1 = RandomReal[{0,1.01},{4,4}];
m2 = RandomReal[{.99,2},{4,4}];

m1\[VectorLess]m2

Compare symmetric matrices:

x = IdentityMatrix[3];y = DiagonalMatrix[{4,5,6}]; z =Dot[#,Transpose[#]]&[RandomReal[1,{3,3}]];

x(Subscript[\[VectorLess], {"SemidefiniteCone",3}])y

PositiveDefiniteMatrixQ[y-x]

x(Subscript[\[VectorLess], {"SemidefiniteCone",3}])z

PositiveDefiniteMatrixQ[z-x]

Represent the condition that (RefLink[Norm, paclet:ref/Norm])[{x,y}]<1:

0(Subscript[\[VectorLess], {"NormCone", 3}]){x,y,1}

% /. {x->.5,y->.6}

Represent the condition that (e^x)<z:

0(Subscript[\[VectorLess], "ExponentialCone"]){x,1,z}

% /. {x->1,z->3}

Show the boundary where ((x)^α)(y^(1-α))>|z| for non-negative x, y with α between 0 and 1:

Manipulate[RegionPlot3D[0(Subscript[\[VectorLess], (PowerConeList[α])]){x,y,z},{x,0,2},{y,0,2},{z,-2,2}],{{α,.5},0.01,.99}]

OpenersExampleSection3(Spacer1)Applications  (1)

(RefLink[VectorLess, paclet:ref/VectorLess]) is a fast way to compare many elements:

v1 = RandomReal[{-1,0},10^7];
v2 = RandomReal[{1,0},10^7];

AbsoluteTiming[v1\[VectorLess]v2]

AbsoluteTiming[AllTrue[v2-v1,Positive]]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Less, paclet:ref/Less])  ▪  (RefLink[VectorLessEqual, paclet:ref/VectorLessEqual])  ▪  (RefLink[VectorGreater, paclet:ref/VectorGreater])  ▪  (RefLink[PositiveReals, paclet:ref/PositiveReals])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Convex Optimization, paclet:guide/ConvexOptimization])}})}})