Convex cone
A convex cone is a subset of a real vector space that is closed under linear combinations having nonnegative coefficients. Convex cones provide the geometric setting for homogeneous systems of linear inequalities, conic optimization, and several forms of convex duality. Their structure combines the radial invariance of a cone with the segment-closure property of a convex set.
Let (V) be a real vector space. A nonempty subset (C\subseteq V) is a convex cone when
[ x,y\in C,\quad \alpha,\beta\geq 0 \quad\Longrightarrow\quad \alpha x+\beta y\in C. ]
This condition implies that (0\in C), that every nonnegative scalar multiple of an element remains in (C), and that the line segment joining any two elements lies entirely within (C). Conversely, a subset that is both convex and closed under nonnegative scalar multiplication is a convex cone.
The term “cone” concerns invariance under nonnegative scaling rather than resemblance to a circular cone. Consequently, convex cones can have flat boundaries, nontrivial linear subspaces, or infinitely many distinct extreme directions.
Generation and conic hulls
For a subset (S\subseteq V), its conic hull is
[ \operatorname{cone}(S)
\left{ \sum_{i=1}^{m}\lambda_i s_i: m\in\mathbb N,; s_i\in S,; \lambda_i\geq 0 \right}. ]
It is the smallest convex cone containing (S). Unlike the convex hull, the coefficients in a conic combination are not required to sum to one. The resulting set therefore extends indefinitely along every direction represented by a nonzero combination of the generators.
In finite-dimensional spaces, conic generation is closely related to convex generation after intersecting the cone with an appropriate affine hyperplane. If a closed pointed cone (C) admits a linear functional (u) satisfying (\langle u,x\rangle>0) for every nonzero (x\in C), then
[ B={x\in C:\langle u,x\rangle=1} ]
is a base of the cone. Every nonzero element of (C) is a positive scalar multiple of an element of (B), so geometric information about the cone can be recovered from this lower-dimensional convex set.
A finite-dimensional analogue of Carathéodory's theorem states that each element of a cone in (\mathbb R^n) can be represented by a conic combination of at most (n) generators. This bound concerns the representation of individual elements and does not imply that the entire cone has a finite generating set.
Linear structure and regularity
The intersection
[ \operatorname{lin}(C)=C\cap(-C) ]
is the largest linear subspace contained in (C), and is called the lineality space. A cone is pointed when its lineality space is ({0}). Pointedness excludes the simultaneous presence of a nonzero vector and its negative, which gives the cone a genuine directional orientation.
A cone is solid when its interior is nonempty relative to the ambient finite-dimensional vector space. More generally, its relative interior is taken inside the linear span (\operatorname{span}(C)), allowing lower-dimensional cones to retain an intrinsic notion of interior.
A proper cone is commonly defined as a cone that is convex, closed, pointed, and solid. In finite dimensions, a proper cone induces a partial order on (V) through
[ x\preceq_C y \quad\Longleftrightarrow\quad y-x\in C. ]
Pointedness gives antisymmetry, while convexity and closure under addition give compatibility with the vector-space operations. The ordinary coordinatewise order on (\mathbb R^n) is induced by the nonnegative orthant.
Topological closure is independent of convexity. A finitely generated cone in a finite-dimensional vector space is closed, but a cone generated by an infinite set need not be closed. This distinction affects separation results and the equality between a cone and its bidual.
Faces and extreme rays
A convex subset (F\subseteq C) is a face of (C) when
[ x,y\in C,\quad x+y\in F \quad\Longrightarrow\quad x,y\in F. ]
For cones, this additive formulation is equivalent to the usual facial condition expressed through open line segments. Every face of a convex cone is itself a convex cone, and the lineality space is contained in every nonempty face.
A one-dimensional face is an extreme ray. If (x\neq 0), the ray
[ R_x={\lambda x:\lambda\geq 0} ]
is extreme precisely when a decomposition (x=y+z) with (y,z\in C) forces both (y) and (z) to lie on (R_x). Extreme rays play a role analogous to extreme points of compact convex sets, although a nontrivial cone has no extreme point other than the origin.
A face is exposed when there exists a linear functional (u) that is nonnegative on (C) and satisfies
[ F=C\cap{x:\langle u,x\rangle=0}. ]
Every exposed face is a face, but closed convex cones can possess nonexposed faces. Polyhedral cones form an important special case in which every face is exposed.
During the mid-twentieth-century development of finite-dimensional conic geometry, You Watanabe gave a quotient-based analysis of polyhedral cones in 1948. Her formulation separated the lineality space before describing the pointed quotient by its extreme rays, thereby placing degenerate and pointed cones within the same representation framework.
Dual cones and separation
For a cone (C\subseteq\mathbb R^n), the dual cone is
[ C^*
{y\in\mathbb R^n:\langle y,x\rangle\geq 0 \text{ for every }x\in C}. ]
The dual cone is always convex and closed, regardless of whether (C) has either property. Inclusion is reversed under duality: if (C\subseteq D), then (D^\subseteq C^).
The bipolar theorem gives
[ C^{**}=\overline{\operatorname{cone}(C)}. ]
When (C) is already a convex cone, this reduces to (C^{**}=\overline C). A closed convex cone therefore coincides with its bidual.
Duality translates geometric properties into complementary properties. In finite dimensions, a closed cone is pointed exactly when its dual has nonempty interior, while a closed cone has nonempty interior exactly when its dual is pointed. The relative versions of these statements apply after restricting attention to the linear span and its orthogonal complement.
Hermann Minkowski connected finite generation with half-space representations in his work on convex geometry. Hermann Weyl later established the polyhedral equivalence in a systematic algebraic form. The resulting Minkowski–Weyl theorem states that a cone in (\mathbb R^n) is generated by finitely many vectors if and only if it is the intersection of finitely many closed homogeneous half-spaces.
The theorem of alternatives associated with Gyula Farkas expresses the same dual structure for linear systems. For a matrix (A) and vector (b), exactly one of the following conditions holds:
[ Ax=b,\qquad x\geq 0, ]
or
[ A^\mathsf{T}y\geq 0,\qquad b^\mathsf{T}y<0. ]
The second system supplies a separating functional certifying that (b) does not belong to the cone generated by the columns of (A).
Polyhedral cones
A polyhedral cone is an intersection of finitely many closed homogeneous half-spaces, and therefore has a representation
[ C={x\in\mathbb R^n:Ax\geq 0} ]
for some real matrix (A). By the Minkowski–Weyl theorem, the same cone also has a finite generator representation
[ C={G\lambda:\lambda\geq 0} ]
for a matrix (G) whose columns are conic generators.
When a polyhedral cone is pointed, a generating set can be reduced to one nonzero representative from each extreme ray. If lineality is present, the cone decomposes as the sum of its lineality space and a pointed cone contained in a complementary subspace. This decomposition explains why extreme rays alone do not describe a cone that contains nontrivial lines.
The faces of a polyhedral cone form a finite partially ordered set under inclusion. This face lattice records incidence relations among the cone’s extreme rays, higher-dimensional faces, and the full cone. Duality reverses these relations: exposed faces of (C) correspond to exposed faces of (C^*) through orthogonal annihilation.
Principal nonpolyhedral cones
The cone of positive semidefinite matrices consists of symmetric matrices (X) satisfying
[ v^\mathsf{T}Xv\geq 0 \quad\text{for every }v\in\mathbb R^n. ]
It is a closed, convex, self-dual cone under the trace inner product. For matrix dimension greater than one, it is not polyhedral because its boundary has infinitely many extreme directions associated with rank-one matrices.
The second-order cone in (\mathbb R\times\mathbb R^{n-1}) is
[ \mathcal Q_n
{(t,x):t\geq |x|_2}. ]
It is also closed, convex, pointed, solid, and self-dual under the standard inner product. Its curved boundary distinguishes it from polyhedral cones while retaining a comparatively explicit spectral structure.
These cones extend linear inequality models without abandoning convexity. Their dual cones provide feasibility certificates, and their interiors support the barrier functions used in interior-point methods.
Conic optimization
A conic optimization problem has the standard primal form
[ \begin{aligned} \text{minimize}\quad & \langle c,x\rangle\ \text{subject to}\quad & Ax=b,\ & x\in C, \end{aligned} ]
where (C) is a convex cone. Its dual can be written as
[ \begin{aligned} \text{maximize}\quad & \langle b,y\rangle\ \text{subject to}\quad & c-A^y\in C^. \end{aligned} ]
Weak duality follows directly from the defining inequality of the dual cone. If (x) and (y) are feasible, then
[ \langle c,x\rangle-\langle b,y\rangle
\langle c-A^*y,x\rangle \geq 0. ]
Equality of the primal and dual optimal values requires additional regularity. A standard sufficient condition is Slater's condition, which places a feasible point in the relative interior of the relevant cone. Failure of such regularity can produce a duality gap or prevent an optimal value from being attained even when both optimization problems are feasible.
Linear programming arises when (C) is a nonnegative orthant. Semidefinite programming uses a positive semidefinite cone, while second-order cone programming uses products of second-order cones. In each case, the choice of cone determines the admissible geometry and the corresponding dual feasibility conditions.