Convex cone

A subset C of a vector space V over an ordered field F is a cone (or sometimes called a linear cone) if for each x in C and positive scalar α in F, the product αx is in C.[1] Note that some authors define cone with the scalar α ranging over all non-negative scalars (rather than all positive scalars, which does not include 0).[2]

A cone C is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.[3][4] A cone C is convex if and only if C + C ⊆ C.

This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real 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 to C.[5] Because of the scaling parameters α and β, cones are infinite in extent and not bounded.

If C is a convex cone, then for any positive scalar α and any x in C the vector ${\displaystyle \alpha x={\tfrac {\alpha }{2}}x+{\tfrac {\alpha }{2}}x\in C.}$ It follows that a convex cone C is a special case of a linear cone.

It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set C is a convex cone if and only if αC = C and C + C = C, for any positive scalar α.

Convex cone that is not a conic combination 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 space V, the empty set, the space V, and any linear subspace of V are convex cones.
• The conical combination of a finite or infinite set of vectors in ${\displaystyle \mathbb {R} ^{n}}$ is a convex cone.
• The tangent cones of a convex set are convex cones.
• The set
  $${\displaystyle \left\{x\in \mathbb {R} ^{2}\mid x_{2}\geq 0,x_{1}=0\right\}\cup \left\{x\in \mathbb {R} ^{2}\mid x_{1}\geq 0,x_{2}=0\right\}}$$

  
  is a cone but not a convex cone.
• The norm cone
  $${\displaystyle \left\{(x,r)\in \mathbb {R} ^{d+1}\mid \|x\|\leq r\right\}}$$

  
  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 arbitrary linear maps. In particular, if C is a convex cone, so is its opposite ${\displaystyle -C}$ and ${\displaystyle C\cap -C}$ is the largest linear subspace contained in C.
• The set of positive semidefinite matrices.
• The set of nonnegative continuous functions is a convex cone.

Let C ⊂ V be a set, not necessary a convex set, in a real vector space V equipped with an inner product. The (continuous or topological) dual cone to C is the set

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

which is always a convex cone. Here, ${\displaystyle \langle w,v\rangle }$ is the duality pairing between C and V, i.e. ${\displaystyle \langle w,v\rangle =v(w)}$.

More generally, the (algebraic) dual cone to C ⊂ V in a linear space V is a subset of the dual space V* defined by:

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

In other words, if V* is the algebraic dual space of V, C* is the set of linear functionals that are nonnegative on the primal cone C. If we take V* to be the continuous dual space then it is the set of continuous linear functionals nonnegative on C.[29] This notion does not require the specification of an inner product on V.

In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous,[30] and every continuous linear functional in an inner product space induces a linear isomorphism (nonsingular linear map) from V* to V, and this isomorphism will take the dual cone given by the second definition, in V*, onto the one given by the first definition; see the Riesz representation theorem.[29]

If C is equal to its dual cone, then C is called self-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.

• Given a closed, convex subset K of Hilbert space V, the outward normal cone to the set K at the point x in K is given by

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

• Given a closed, convex subset K of V, the tangent cone (or contingent cone) to the set K at the point x is given by

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

• Given a closed, convex subset K of Hilbert space V, the tangent cone to the set K at the point x in K can be defined as polar cone to outwards normal cone ${\displaystyle N_{K}(x)}$:

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

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

They are important concepts in the fields of convex optimization, variational inequalities and projected dynamical systems.

If C is a non-empty convex cone in X, then the linear span of C is equal to C - C and the largest vector subspace of X contained in C is equal to C ∩ (−C).[31]

A pointed and salient convex cone C induces a partial ordering "≤" on V, defined so that ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in C.}$ (If the cone is flat, the same definition gives merely a preorder.) 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 an ordered vector space. Examples include the product order on real-valued vectors, ${\displaystyle \mathbb {R} ^{n},}$ and the Loewner order on positive semidefinite matrices. Such an ordering is commonly found in positive semidefinite programming.

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

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• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 1-4008-7317-7.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. ISBN 981-238-067-1. MR 1921556.