Convex cone

Inlinear algebra,acone—sometimes called alinear conefor distinguishing it from other sorts of cones—is a subset of avector spacethat isclosedunder positive scalar multiplication; that is,Cis a cone if$x\in C$implies$sx\in C$for everypositive scalars.A cone need not be convex, or even look like aconeinEuclidean space.

A convex cone (light blue). Inside of it, the light red convex cone consists of all pointsαx+βywithα, β > 0,for the depictedxandy.The curves on the upper right symbolize that the regions are infinite in extent.

When the scalars are real numbers, or belong to anordered field,one generally calls aconea subset of a vector space that is closed under multiplication by apositive scalar.In this context, aconvex coneis a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed underlinear combinationswith positive coefficients. It follows that convex cones areconvex sets.

In this article, only the case of scalars in an ordered field is considered.

Definition

AsubsetCof a vector spaceVover anordered fieldFis acone(or sometimes called alinear cone) if for eachxinCand positive scalarαinF,the productαxis inC.Note that some authors defineconewith the scalarαranging over all non-negative scalars (rather than all positive scalars, which does not include 0).

A coneCis aconvex coneifαx+βybelongs toC,for any positive scalarsα,β,and anyx,yinC. A coneCis convex if and only ifC+C⊆C.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over therational,algebraic,or (more commonly) thereal numbers.Also note that the scalars in the definition are positive meaning that the origin does not have to belong to C. Some authors use a definition that ensures the origin belongs toC.Because of the scaling parametersαandβ,cones are infinite in extent and not bounded.

IfCis a convex cone, then for any positive scalarαand anyxinCthe vector$\alpha x={\tfrac {\alpha }{2}}x+{\tfrac {\alpha }{2}}x\in C.$It follows that a convex coneCis a special case of alinear cone.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed underconvex combinations,or just underadditions.More succinctly, a setCis a convex cone if and only ifαC=CandC+C=C,for any positive scalarα.

Examples

Convex cone that is not a conic hull of finitely many generators.

Convex cone generated by the conic combination of the three black vectors.

A cone (the union of two rays) that is not a convex cone.

• For a vector spaceV,the empty set, the spaceV,and anylinear subspaceofVare convex cones.
• Theconical hullof a finite or infinite set of vectors in$\mathbb {R} ^{n}$is a convex cone.
• Thetangent conesof a convex set are convex cones.
• The set$$\left\{x\in \mathbb {R} ^{2}\mid x_{2}\geq 0,x_{1}=0\right\}\cup \left\{x\in \mathbb {R} ^{2}\mid x_{1}\geq 0,x_{2}=0\right\}$$is a cone but not a convex cone.
• The norm cone$$\left\{(x,r)\in \mathbb {R} ^{d+1}\mid \|x\|\leq r\right\}$$is a convex cone.
• The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one.
• The class of convex cones is also closed under arbitrarylinear maps.In particular, ifCis a convex cone, so is its opposite$-C$and$C\cap -C$is the largest linear subspace contained inC.
• The set ofpositive semidefinite matrices.
• The set of nonnegative continuous functions is a convex cone.

Special examples

Affine convex cones

Anaffine convex coneis the set resulting from applying an affine transformation to a convex cone.A common example is translating a convex cone by a pointp:p+C.Technically, such transformations can produce non-cones. For example, unlessp= 0,p+Cis not a linear cone. However, it is still called an affine convex cone.

Half-spaces

A (linear)hyperplaneis a set in the form$\{x\in V\mid f(x)=c\}$where f is alinear functionalon the vector space V. Aclosedhalf-spaceis a set in the form$\{x\in V\mid f(x)\leq c\}$or$\{x\in V\mid f(x)\geq c\},$and likewise an open half-space uses strict inequality.

Half-spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex coneCthat is not the whole spaceVmust be contained in some closed half-spaceHofV;this is a special case ofFarkas' lemma.

Polyhedral and finitely generated cones

Polyhedral conesare special kinds of cones that can be defined in several ways:

• A coneCis polyhedral if it is theconical hullof finitely many vectors (this property is also calledfinitely-generated).I.e., there is a set of vectors$\{v_{1},\ldots,v_{k}\}$so that$C=\{a_{1}v_{1}+\cdots +a_{k}v_{k}\mid a_{i}\in \mathbb {R} _{\geq 0},v_{i}\in \mathbb {R} ^{n}\}$.
• A cone is polyhedral if it is the intersection of a finite number of half-spaces which have 0 on their boundary (the equivalence between these first two definitions was proved by Weyl in 1935).
• A coneCis polyhedral if there is somematrix$A$such that$C=\{x\in \mathbb {R} ^{n}\mid Ax\geq 0\}$.
• A cone is polyhedral if it is the solution set of a system of homogeneous linear inequalities. Algebraically, each inequality is defined by a row of the matrixA.Geometrically, each inequality defines a halfspace that passes through the origin.

Every finitely generated cone is a polyhedral cone, and every polyhedral cone is a finitely generated cone.Every polyhedral cone has a unique representation as a conical hull of its extremal generators, and a unique representation of intersections of halfspaces, given each linear form associated with the halfspaces also define a support hyperplane of a facet.

Polyhedral cones play a central role in the representation theory ofpolyhedra.For instance, the decomposition theorem for polyhedra states that every polyhedron can be written as theMinkowski sumof aconvex polytopeand a polyhedral cone.Polyhedral cones also play an important part in proving the relatedFinite Basis Theoremfor polytopes which shows that every polytope is a polyhedron and everyboundedpolyhedron is a polytope.

The two representations of a polyhedral cone - by inequalities and by vectors - may have very different sizes. For example, consider the cone of all non-negativen-by-nmatrices with equal row and column sums. The inequality representation requiresn²inequalities and2(n− 1)equations, but the vector representation requiresn!vectors (see theBirkhoff-von Neumann Theorem). The opposite can also happen - the number of vectors may be polynomial while the number of inequalities is exponential.

The two representations together provide an efficient way to decide whether a given vector is in the cone: to show that it is in the cone, it is sufficient to present it as a conic combination of the defining vectors; to show that it is not in the cone, it is sufficient to present a single defining inequality that it violates. This fact is known asFarkas' lemma.

A subtle point in the representation by vectors is that the number of vectors may be exponential in the dimension, so the proof that a vector is in the cone might be exponentially long. Fortunately,Carathéodory's theoremguarantees that every vector in the cone can be represented by at mostddefining vectors, wheredis the dimension of the space.

Blunt, pointed, flat, salient, and proper cones

According to the above definition, ifCis a convex cone, thenC∪ {0} is a convex cone, too. A convex cone is said to bepointedif0is inC,andbluntif0is not inC.Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β.

A cone is calledflatif it contains some nonzero vectorxand its opposite −x,meaningCcontains a linear subspace of dimension at least one, andsalientotherwise. A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex coneCis salient if and only ifC∩ −C⊆ {0}. A coneCis said to begeneratingif$C-C=\{x-y\mid x\in C,y\in C\}$equals the whole vector space.

Some authors require salient cones to be pointed. The term "pointed" is also often used to refer to a closed cone that contains no complete line (i.e., no nontrivial subspace of the ambient vector spaceV,or what is called a salient cone). The termproper(convex)coneis variously defined, depending on the context and author. It often means a cone that satisfies other properties like being convex, closed, pointed, salient, and full-dimensional.Because of these varying definitions, the context or source should be consulted for the definition of these terms.

Rational cones

A type of cone of particular interest to pure mathematicians is thepartially ordered setof rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming.".This object arises when we study cones in$\mathbb {R} ^{d}$together with thelattice$\mathbb {Z} ^{d}$.A cone is calledrational(here we assume "pointed", as defined above) whenever its generators all haveintegercoordinates, i.e., if$C$is a rational cone, then$C=\{a_{1}v_{1}+\cdots +a_{k}v_{k}\mid a_{i}\in \mathbb {R} _{+}\}$for some$v_{i}\in \mathbb {Z} ^{d}$.

Dual cone

Main article:dual cone

LetC⊂Vbe a set, not necessarily a convex set, in a real vector spaceVequipped with aninner product.The (continuous or topological)dual conetoCis the set

$C^{*}=\{v\in V\mid \forall w\in C,\langle w,v\rangle \geq 0\},$

which is always a convex cone. Here,$\langle w,v\rangle$is theduality pairingbetweenCandV,i.e.$\langle w,v\rangle =v(w)$.

More generally, the (algebraic) dual cone toC⊂Vin a linear space V is a subset of thedual spaceV*defined by:

$C^{*}:=\left\{v\in V^{*}\mid \forall w\in C,v(w)\geq 0\right\}.$

In other words, ifV*is thealgebraic dual spaceofV,C*is the set of linear functionals that are nonnegative on the primal coneC.If we takeV*to be thecontinuous dual spacethen it is the set of continuous linear functionals nonnegative onC.This notion does not require the specification of an inner product onV.

In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous,and every continuous linear functional in an inner product space induces a linear isomorphism (nonsingular linear map) fromV*toV,and this isomorphism will take the dual cone given by the second definition, inV*,onto the one given by the first definition; see theRiesz representation theorem.

IfCis equal to its dual cone, thenCis calledself-dual.A cone can be said to be self-dual without reference to any given inner product, if there exists an inner product with respect to which it is equal to its dual by the first definition.

Constructions

• Given a closed, convex subsetKofHilbert spaceV,theoutward normal coneto the setKat the pointxinKis given by

  $N_{K}(x)=\left\{p\in V\colon \forall x^{*}\in K,\left\langle p,x^{*}-x\right\rangle \leq 0\right\}.$

• Given a closed, convex subsetKofV,thetangent cone(orcontingent cone) to the setKat the pointxis given by

  $T_{K}(x)={\overline {\bigcup _{h>0}{\frac {K-x}{h}}}}.$

• Given a closed, convex subsetKof Hilbert spaceV,thetangent coneto the setKat the pointxinKcan be defined aspolar coneto outwards normal cone$N_{K}(x)$:

  $T_{K}(x)=N_{K}^{o}(x)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{y\in V\mid \forall \xi \in N_{K}(x):\langle y,\xi \rangle \leqslant 0\}$

Both the normal and tangent cone have the property of being closed and convex.

They are important concepts in the fields ofconvex optimization,variational inequalitiesandprojected dynamical systems.

Properties

IfCis a non-empty convex cone inX,then the linear span ofCis equal toC-Cand the largest vector subspace ofXcontained inCis equal toC∩ (−C).

Partial order defined by a convex cone

A pointed and salient convex coneCinduces apartial ordering"≥" onV,defined so that$x\geq y$if and only if$x-y\in C.$(If the cone is flat, the same definition gives merely apreorder.) Sums and positive scalar multiples of valid inequalities with respect to this order remain valid inequalities. A vector space with such an order is called anordered vector space.Examples include theproduct orderon real-valued vectors,$\mathbb {R} ^{n},$and theLoewner orderon positive semidefinite matrices. Such an ordering is commonly found insemidefinite programming.

See also

• Cone (disambiguation)
  • Cone (geometry)
  • Cone (topology)
• Farkas' lemma
• Bipolar theorem
• Ordered vector space

References

• Bourbaki, Nicolas(1987).Topological Vector Spaces.Elements of Mathematics. Berlin, New York:Springer-Verlag.ISBN978-3-540-13627-9.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces.Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN978-1584888666.OCLC144216834.
• Rockafellar, R. T.(1997) [1970].Convex Analysis.Princeton, NJ: Princeton University Press.ISBN1-4008-7317-7.
• Schaefer, Helmut H.;Wolff, Manfred P. (1999).Topological Vector Spaces.GTM.Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN978-1-4612-7155-0.OCLC840278135.
• Trèves, François(2006) [1967].Topological Vector Spaces, Distributions and Kernels.Mineola, N.Y.: Dover Publications.ISBN978-0-486-45352-1.OCLC853623322.
• Zălinescu, C. (2002).Convex Analysis in General Vector Spaces.River Edge, NJ: World Scientific.ISBN981-238-067-1.MR1921556.