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Convex cone

A convex cone is a subset C of a real vector space that is both convex and closed under nonnegative scalar multiplication, meaning that for any x, y \in C and any \theta, \lambda \geq 0, the combination \theta x + \lambda y also belongs to C.^[1] This structure captures sets invariant under positive scaling and addition while preserving convexity, distinguishing convex cones from more general cones that may lack convexity.^[1] Convex cones form a foundational concept in convex analysis and optimization, enabling the formulation of conic programs where constraints are expressed via membership in such sets.^[1] Notable examples include the nonnegative orthant \mathbb{R}^n_+, defined as \{x \in \mathbb{R}^n \mid x \geq 0\} componentwise, which models nonnegativity constraints in linear programming.^[1] Another key example is the second-order cone (or Lorentz cone), given by \{(x, t) \in \mathbb{R}^n \times \mathbb{R} \mid \|x\|_2 \leq t, t \geq 0\}, which arises in robust optimization and quadratic constraints.^[1] The positive semidefinite cone S^n_+, consisting of symmetric n \times n matrices X such that X \succeq 0, is central to semidefinite programming and applications in control theory and machine learning.^[1] Key properties of convex cones include closure under arbitrary intersections, ensuring that the intersection of any family of convex cones remains a convex cone.^[1] The dual cone C^* = \{y \mid y^T x \geq 0 \ \forall x \in C\} is itself a convex cone, facilitating duality theory in optimization where primal and dual problems are linked through cone structures.^[1] A convex cone is termed proper if it is convex, closed, full-dimensional (with nonempty interior), and pointed (containing no line through the origin), which is essential for defining generalized inequalities like x \preceq_K y if y - x \in K.^[1] In practice, convex cones underpin efficient algorithms such as interior-point methods for solving large-scale problems in engineering, finance, and signal processing, where they ensure global optimality and computational tractability.^[1] Their role extends to generalized inequalities, allowing extensions of linear and semidefinite programming to broader classes of problems, including those involving norms and spectrahedra.^[1]

Definitions and Basic Concepts

General Cone

In a real vector space V, a cone is defined as a nonempty subset C \subseteq V that is closed under nonnegative scalar multiplication, meaning that for every x \in C and every \lambda \geq 0, it holds that \lambda x \in C.^[2] This property ensures that C contains all rays emanating from the origin through its points, including the origin itself, since taking \lambda = 0 yields $0 \in C.^[3] Alternative definitions exist, particularly in some geometric and early abstract contexts, where a cone is specified using strictly positive scalars \lambda > 0, without explicitly requiring the origin.^[4] In such formulations, the set consists of half-lines (rays) starting from but not necessarily including the origin, and the origin lies in the closure of the cone if it is nonempty. These ray-based definitions emphasize the directional aspect but can complicate algebraic operations. Modern treatments in functional analysis and optimization typically adopt the inclusive definition with \lambda \geq 0 to align with structures like ordered vector spaces and to ensure the origin is an interior point for certain properties.^[5] This inclusion facilitates compatibility with broader set operations, such as those involving convex sets, where the zero vector serves as a natural boundary element. Historically, the notion traces back to 19th-century projective geometry, where cones were viewed as unions of lines from a vertex, often excluding the vertex itself; the shift to including the origin became standard in 20th-century functional analysis to support duality and ordering theories.^[6]

Convex Cone

In real vector spaces over the field of real numbers \mathbb{R}, a convex cone C \subseteq V is defined as a subset that is both a cone—closed under nonnegative scalar multiplication—and convex, meaning that for all x, y \in C and \lambda, \mu \geq 0, the linear combination \lambda x + \mu y \in C.^[1] This property ensures that C is closed under nonnegative linear combinations of its elements.^[7] Equivalently, C contains all finite sums of the form \sum_{i=1}^k \lambda_i x_i, where k \in \mathbb{N}, \lambda_i \geq 0, and x_i \in C for each i.^[1] Building on the scaling property of general cones, the addition of convexity in this definition implies that convex cones are stable under summation, distinguishing them from non-convex cones that may lack this closure.^[7] In contrast to general convex sets, which can be bounded (such as closed balls), convex cones exhibit unboundedness along any ray from the origin through an element of the cone, as \lambda x \in C for all \lambda \geq 0 and x \in C, though they may be bounded in other directions.^[1] Although the definition applies to arbitrary real vector spaces, finite-dimensional cases like \mathbb{R}^n provide key intuition, where geometric visualizations highlight the cone's structure as a collection of rays filling a convex region from the origin.^[7]

Exposed Faces and Faces

In convex analysis, a face of a convex cone C is defined as a convex subset F \subseteq C such that whenever a point in F is expressed as a convex combination of points from C, all those points must also belong to F. Formally, if y = \sum_{i=1}^k \lambda_i x_i with \lambda_i \geq 0, \sum_{i=1}^k \lambda_i = 1, x_i \in C, and y \in F, then x_i \in F for all i. This property ensures that faces are the "extreme" subcones preserved under the convex combinations and positive scalings inherent to cone operations.^[7] Exposed faces represent a geometrically distinct class of faces, obtained by intersecting the cone with a supporting hyperplane. A subset F \subseteq C is an exposed face if there exists a supporting hyperplane H to C such that F = C \cap H. For convex cones, supporting hyperplanes necessarily pass through the origin, so the form simplifies to F = \{ x \in C \mid \langle a, x \rangle = 0 \}, where a is a nonzero linear functional satisfying \langle a, y \rangle \geq 0 for all y \in C. Every exposed face is itself a face, but the converse does not hold in general.^[8] While polyhedral convex cones in finite dimensions are always facially exposed—meaning every face is exposed—the structure of faces in general convex cones can differ, even in finite dimensions, with counterexamples of non-exposed faces existing. In infinite-dimensional settings, non-exposed faces are more prevalent due to topological complexities, contrasting with the finite-dimensional case where stricter regularity often applies. The Krein-Milman theorem underscores this distinction by guaranteeing that a compact convex set equals the closed convex hull of its extreme points in any dimension, but requiring closure in infinite dimensions, which parallels challenges in exposing faces of cones.^[7]^[9]

Examples and Special Cases

Non-Polyhedral Examples

Non-polyhedral convex cones often arise from infinite dimensionality or nonlinear constraints. A natural example in infinite dimensions is the positive cone in \ell_p spaces for $1 \leq p \leq \infty, comprising all sequences x = (x_i)_{i=1}^\infty with x_i \geq 0 for all i and \sum_{i=1}^\infty |x_i|^p < \infty (or the sup norm finite for p = \infty); this forms a closed convex cone that lacks a finite generating set of extreme rays due to the infinite dimensionality.^[10] Another prominent non-polyhedral example is the Lorentz cone, also called the second-order cone or ice-cream cone, defined in \mathbb{R}^{n+1} as \mathcal{L}^{n+1} = \{ (x, t) \in \mathbb{R}^n \times \mathbb{R} \mid \|x\|_2 \leq t \}.^[10] This set is a closed convex cone with a smooth, curved boundary arising from the quadratic constraint, distinguishing it from polyhedral cones that rely solely on linear facets; it is self-dual and plays a central role in second-order cone programming for applications like robust optimization.^[10] The cone of positive semidefinite matrices provides yet another key non-polyhedral instance, given by S_+^n = \{ X \in S^n \mid X \succeq 0 \}, where S^n denotes the space of n \times n real symmetric matrices and X \succeq 0 if all eigenvalues of X are nonnegative (equivalently, z^T X z \geq 0 for all z \in \mathbb{R}^n).^[10] As a closed convex cone, it is non-polyhedral because its definition requires infinitely many linear inequalities, leading to a curved structure in the space of matrices; this cone is self-dual and essential in semidefinite programming for problems in control theory and combinatorial optimization.^[10] These examples—the infinite-dimensional positive orthant analogs, the Lorentz cone, and the positive semidefinite cone—demonstrate closed convex cones that are non-polyhedral primarily due to curvature from nonlinear boundaries or the absence of finite extremal structure in infinite dimensions.^[10]

Polyhedral Cones

A polyhedral cone is a convex cone in finite-dimensional space that arises as the intersection of finitely many closed half-spaces, each containing the origin in its boundary. Formally, for a matrix A \in \mathbb{R}^{m \times n} with m < \infty, the set C = \{ x \in \mathbb{R}^n \mid A x \geq 0 \} defines a polyhedral cone, where the inequalities correspond to the supporting hyperplanes of the half-spaces.^[11] This H-representation (inequality description) captures the faceted, piecewise-linear structure of the cone, distinguishing it from more general convex cones that may require infinitely many constraints.^[12] An illustrative example in \mathbb{R}^3 is the non-negative orthant, a simplicial polyhedral cone defined by the intersection of three half-spaces: x_1 \geq 0, x_2 \geq 0, and x_3 \geq 0. To demonstrate the effect of an additional half-space, consider the cone C = \{ x \in \mathbb{R}^3 \mid x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0, \, x_1 + 2x_2 + x_3 \geq 0 \}; here, the fourth inequality is redundant within the orthant but highlights how finite intersections refine the cone while preserving polyhedrality.^[11] Such cones appear in applications like linear programming, where they model feasible regions bounded by linear constraints.^[13] Farkas' lemma establishes solvability conditions for linear systems involving polyhedral cones, stating that the system A x = b, x \geq 0 has a solution if and only if every y satisfying A^T y \geq 0 also satisfies y^T b \geq 0. In the context of polyhedral cones, this lemma provides an alternative characterization: a vector lies outside the cone \{ x \mid A x \geq 0 \} if there exists a separating hyperplane defined by a non-negative combination of the rows of A.^[13] This duality underpins algorithms for checking membership and optimizing over such cones.^[14] In finite dimensions, the Weyl-Minkowski theorem asserts that every polyhedral cone is finitely generated, meaning it equals the set of non-negative linear combinations of a finite set of vectors.^[15] This equivalence between the H-representation and the V-representation (generator description) is foundational for computational geometry and optimization, though the proof relies on advanced techniques like Fourier-Motzkin elimination.^[16]

Finitely Generated Cones

A finitely generated convex cone in a finite-dimensional real vector space is defined as the set of all nonnegative linear combinations of a finite collection of vectors. Specifically, given vectors v_1, \dots, v_m \in \mathbb{R}^n, the cone C = \cone\{v_1, \dots, v_m\} consists of all points of the form \sum_{i=1}^m \lambda_i v_i where \lambda_i \geq 0 for each i.^[13]^[17] This representation, known as the V-representation or ray generation, captures the cone as the conic hull of its generating vectors. The explicit form of the sum emphasizes that the cone is built from rays emanating from the origin along the directions of the generators, with the minimal such set corresponding to the extreme rays of the cone. For a pointed finitely generated cone, the extreme rays provide a unique minimal set of generators, up to positive scaling.^[18] A simple example in \mathbb{R}^2 is the cone generated by the vectors (1,0) and (0,1), which forms the nonnegative first quadrant \{ (x,y) \mid x \geq 0, y \geq 0 \}. This illustrates how finite ray generators produce a closed convex region bounded by the coordinate axes. In finite dimensions, a convex cone is finitely generated if and only if it is polyhedral, by the Minkowski–Weyl theorem.^[13]

Classifications of Convex Cones

Pointed, Blunt, and Lineal Cones

A convex cone C in a vector space is classified based on its lineality space, defined as \operatorname{lin}(C) = C \cap (-C), which is the largest linear subspace contained in C. This space consists of all directions x such that both x and -x belong to C, or equivalently, \operatorname{lin}(C) = \{x \mid x, -x \in C\}. The lineality space captures the "linear" component of the cone, representing directions in which the cone extends infinitely in both positive and negative senses from the origin. A convex cone C is pointed if its lineality space is trivial, i.e., \operatorname{lin}(C) = \{0\}, meaning C contains no complete line through the origin other than the origin itself. Equivalently, if x \in C and -x \in C, then x = 0. This property ensures that the origin is an extreme point of C. A classic example is the nonnegative orthant \mathbb{R}^n_+, where \operatorname{lin}(\mathbb{R}^n_+) = \{0\}, as no nonzero vector and its negative both have nonnegative components. In contrast, a convex cone is blunt if \operatorname{lin}(C) \neq \{0\}, indicating that it contains at least one line through the origin. For instance, the full Euclidean space \mathbb{R}^n is blunt, with \operatorname{lin}(\mathbb{R}^n) = \mathbb{R}^n, as it includes all directions and their opposites. Blunt cones arise naturally when the cone incorporates bidirectional extents, such as in certain recession analyses or when modeling unbounded feasible regions in optimization. A special case of a blunt cone is a lineal cone, where C = \operatorname{lin}(C), meaning the cone is itself a linear subspace (i.e., C = -C). In this decomposition, every convex cone C can be uniquely expressed as C = C' + \operatorname{lin}(C), where C' is a pointed convex cone satisfying C' \cap \operatorname{lin}(C) = \{0\}. This direct sum structure separates the pointed "wedge-like" part from the linear subspace, facilitating analysis in applications like duality and facial decompositions.

Salient, Proper, and Flat Cones

A salient convex cone is defined as a cone C such that for every nonzero x \in C, -x \notin C, or equivalently, C \cap (-C) = \{0\}, meaning it contains no complete line through the origin.^[19] This property ensures that the recession cone of C, which coincides with C itself for cones, is pointed.^[20] In contrast, a flat convex cone contains at least one complete line through the origin, implying a nontrivial lineality space.^[19] A proper convex cone is one that is pointed, closed, and solid, possessing a nonempty interior that forms a solid angle at the origin.^[21] Thus, proper cones are pointed by construction, distinguishing them from more general classifications.^[3] The nonnegative orthant \mathbb{R}^n_+\ = \{ x \in \mathbb{R}^n \mid x_i \geq 0 \ \forall i \} exemplifies a proper cone, as it satisfies all these properties in Euclidean space.^[20] In conic optimization, proper cones play a key role by enabling strict feasibility conditions, where there exists a point in the interior of the cone satisfying the problem constraints, which facilitates strong duality and interior-point methods.^[21] Closed half-spaces through the origin, such as \{ x \in \mathbb{R}^n \mid a^\top x \geq 0 \} for some nonzero a \in \mathbb{R}^n, provide examples of flat cones when the dimension exceeds one, as their lineality space is the hyperplane kernel of a.^[20] Geometrically, proper cones can be visualized as wedge-shaped regions emanating from the origin in a single directional "half," excluding any parallel lines extending in opposite directions, unlike flat cones that incorporate bidirectional subspaces.^[3]

Rational and Polyhedral Variants

A rational polyhedral cone is a convex polyhedral cone in \mathbb{R}^n that admits a description using rational data, either as the conical hull of finitely many rational vectors (finitely generated form) or as the intersection of finitely many half-spaces defined by rational inequalities (inequality form).^[22]^[23] This rationality ensures that the cone's extreme rays or facets can be represented with rational coordinates, facilitating computational and algebraic analysis.^[24] In integer programming, rational polyhedral cones play a central role, as their integer points form a monoid that can be finitely generated, allowing for the resolution of optimization problems over discrete structures.^[22] For instance, consider the cone in \mathbb{R}^2 generated by the integer vectors (1,0) and (1,1); this is a rational polyhedral cone consisting of all points (x,y) satisfying x \geq y \geq 0 and x \geq 0, whose integer points include vectors like (0,0), (1,0), (1,1), and (2,1).^[25] A key property of rational polyhedral cones is the existence of a Hilbert basis: a finite minimal set of integer vectors such that every integer point in the cone is a non-negative integer linear combination of these vectors.^[26] This basis provides the irreducible generators of the monoid of lattice points within the cone and is unique up to the cone's lineality space.^[25] Rational polyhedral cones also connect to algebraic geometry through toric varieties, where each such cone defines an affine toric variety whose coordinate ring is generated by the semigroup of lattice points in the dual cone, as established by Gordan's lemma ensuring finite generation.^[27] Recent computational advances post-2020 in enumerating structures of rational cones, such as Hilbert bases and triangulations, have been driven by enhancements in the Normaliz software, including improved parallel algorithms and support for higher-dimensional instances up to dimension 20 or more. As of 2024, Normaliz version 3.15 further improves performance for dimensions exceeding 20.^[28]^[29]

Dual and Polar Cones

Definition of Dual Cone

In convex analysis, the dual cone of a convex cone C \subseteq V, where V is a real vector space, is defined as the set C^* = \{ y \in V^* \mid \langle y, x \rangle \geq 0 \ \forall x \in C \}, with V^* denoting the algebraic dual space of V.^[30] This construction captures all linear functionals that are nonnegative on every element of C, thereby establishing a duality between the cone and the functionals it "supports" nonnegatively.^[10] In the Euclidean setting, where V is equipped with an inner product and V^* can be identified with V itself via the Riesz representation theorem, the dual cone simplifies to C^* = \{ y \in V \mid \langle y, x \rangle \geq 0 \ \forall x \in C \}.^[10] A related concept is the polar cone, denoted C^\circ = \{ y \in V \mid \langle y, x \rangle \leq 0 \ \forall x \in C \}, which is the negative of the dual cone, C^\circ = -C^*.^[10] This equivalence highlights how the dual cone aligns with supporting hyperplanes that bound C from below. A prominent example is the nonnegative orthant \mathbb{R}^n_+ = \{ x \in \mathbb{R}^n \mid x_i \geq 0 \ \forall i = 1, \dots, n \}, whose dual cone is itself, rendering it self-dual: (\mathbb{R}^n_+)^* = \mathbb{R}^n_+.^[10] Indeed, for any y \in \mathbb{R}^n_+ and x \in \mathbb{R}^n_+, the inner product y^\top x = \sum_{i=1}^n y_i x_i \geq 0, and conversely, any y satisfying this for all x \in \mathbb{R}^n_+ must have nonnegative components.^[10] For closed convex cones, the bidual cone satisfies C^{**} = C, as established by the bipolar theorem, which asserts that taking the dual twice recovers the original cone when it is closed.^[10] More generally, for any convex cone C, the bidual equals the closure \operatorname{cl}(C).^[10] This result underscores the robustness of the duality framework in preserving the structure of convex cones under closure operations.

Properties of Dual Cones

The dual cone C^* of a cone C in a finite-dimensional real vector space is always closed and convex, regardless of whether C itself is convex or closed. This follows from the definition C^* = \{ y \in V^* \mid \langle y, x \rangle \geq 0 \ \forall x \in C \}, as it represents the intersection of closed half-spaces defined by the inequalities \langle y, x \rangle \geq 0. Moreover, if C is a closed convex cone, then the bidual C^{**} = C, establishing a one-to-one correspondence between closed convex cones and their duals under double duality. A key structural property involves the duality between faces of C and faces of C^*. The faces of C correspond bijectively to the faces of C^* through the annihilator mapping, where for a face F of C, the annihilator is defined as \operatorname{ann}(F) = \{ y \in V^* \mid \langle y, x \rangle = 0 \ \forall x \in F \}. Specifically, the dual cone of the face F satisfies (F)^* = \operatorname{cl}(C^* + \operatorname{ann}(F)), where the closure accounts for potential non-closure in infinite-dimensional settings, though in finite dimensions it simplifies to C^* + \operatorname{ann}(F).^[31] This relation highlights how the orthogonal complement to the span of F interacts with C^* to expose the dual facial structure. Self-dual cones, where C^* = C under the given inner product, exhibit symmetric properties under duality and arise in various contexts. Prominent examples include the nonnegative orthant \mathbb{R}_+^n, the Lorentz cone (also known as the second-order cone) \mathcal{L}^{n+1} = \{ (x,t) \in \mathbb{R}^n \times \mathbb{R} \mid \|x\|_2 \leq t \}, and the cone of positive semidefinite matrices \mathcal{S}_+^n.^[10] In infinite-dimensional spaces, the facial structure of dual cones requires additional care due to potential failures of closure and exposure properties. Recent advancements in functional analysis have characterized conditions under which dual cones preserve facial exposure; for instance, a closed convex cone is facially dual complete (or "nice") if C^* + F^\perp is closed for every face F of C, ensuring robust duality in optimization and variational problems. Furthermore, studies on amenable cones demonstrate that such structures admit projectionally exposed facets, facilitating analysis of boundary behavior in spaces like Hilbert or Banach lattices. These results extend classical finite-dimensional theory to infinite dimensions, addressing gaps in earlier frameworks.^[32]

Constructions and Operations

Generation and Minkowski Operations

Convex cones can be constructed from simpler sets using various operations, including Minkowski sums, intersections, and hull operations, which preserve the convexity and conical properties. These constructions allow for building more complex cones from basic building blocks such as rays, half-spaces, or finite sets of vectors. The Minkowski sum of two sets C and D in a vector space is defined as C + D = \{ c + d \mid c \in C, \, d \in D \}. If C and D are convex cones, then C + D is also a convex cone, as it is closed under nonnegative scaling and addition: for any \theta \geq 0 and elements c_1, c_2 \in C, d_1, d_2 \in D, \theta (c_1 + d_1) + (c_2 + d_2) = (\theta c_1 + c_2) + (\theta d_1 + d_2) \in C + D. This operation is particularly useful for decomposing cones into sums of simpler components, such as combining a polyhedral cone with a quadratic cone. The intersection of two convex cones C \cap D is itself a convex cone, since it inherits closure under nonnegative scaling and addition from both C and D. For instance, polyhedral cones are often formed as intersections of finitely many homogeneous half-spaces \{ x \mid \langle a_i, x \rangle \geq 0 \}, where each half-space is a basic convex cone.^[33] A key way to generate convex cones from a set S is through its conic hull, which involves nonnegative linear combinations. The nonnegative hull of S, denoted \mathrm{cone}(S), is the set \{ \sum \lambda_i s_i \mid \lambda_i \geq 0, \, s_i \in S, \, \text{finite sum} \}, forming the smallest convex cone containing S. In contrast, the positive hull \mathrm{pos}(S) uses strictly positive coefficients: \mathrm{pos}(S) = \{ \sum \lambda_i s_i \mid \lambda_i > 0, \, s_i \in S, \, \text{finite sum} \}, which generates the "interior" directions of the cone but excludes the origin unless S contains it.^[34] The closure of \mathrm{pos}(S) often yields the full convex cone when S spans the space. Any closed convex cone can be represented as the intersection of all closed homogeneous half-spaces containing it, dual to the finite generation by extreme rays. This representation underscores the H-description of cones, complementary to the V-description via generators.^[35]

Projection and Sections

The projection of a convex cone C \subseteq \mathbb{R}^n onto a linear subspace U \subseteq \mathbb{R}^n is defined as \operatorname{proj}_U(C) = \{ \operatorname{proj}_U(x) \mid x \in C \}, where \operatorname{proj}_U denotes the orthogonal projection operator onto U. Since the orthogonal projection is a linear map and the image of a convex set under a linear map is convex, \operatorname{proj}_U(C) is convex.^[36] Furthermore, as C is closed under nonnegative scalar multiplication and linear maps preserve such conic combinations, \operatorname{proj}_U(C) is also a convex cone.^[36] A section of a convex cone C is obtained by intersecting C with a linear hyperplane H = \ker f = \{ x \in \mathbb{R}^n \mid f(x) = 0 \}, where f: \mathbb{R}^n \to \mathbb{R} is a linear functional such that f(x) \geq 0 for all x \in C (i.e., H is a supporting hyperplane to C at the origin). The intersection C \cap H is convex, as the intersection of convex sets is convex.^[36] Moreover, since C contains the origin and is a cone, and H passes through the origin, C \cap H inherits the conic structure: for any y \in C \cap H and \lambda \geq 0, \lambda y \in C and \lambda y \in H, so C \cap H is a convex cone.^[37] In fact, such sections correspond to exposed faces of C, which are subcones exposing the facial structure of C.^[37] Projections and sections provide methods to construct lower-dimensional convex cones from higher-dimensional ones, often reducing complexity in analysis or computation. For instance, the linear image of the cone of positive semidefinite matrices under the map extracting the diagonal entries is the nonnegative orthant \mathbb{R}^n_+, as every positive semidefinite matrix has nonnegative diagonal entries, and any nonnegative vector on the diagonal arises from a diagonal positive semidefinite matrix.^[36] This illustrates how projections can simplify cone representations while preserving essential convexity and conic properties. In the polyhedral case, where C = \{ x \mid Ax \geq 0 \} for some matrix A, the projection onto a coordinate subspace (eliminating certain variables) yields another polyhedral cone, computable via Fourier-Motzkin elimination, which systematically generates the inequalities defining the projected cone.^[38] This process preserves the finiteness of the inequality description, relating directly to the facial structure by revealing exposed faces through variable elimination.^[38]

Structural Properties

Extreme Rays and Generators

In convex analysis, an extreme ray of a convex cone C \subseteq \mathbb{R}^n is defined as a one-dimensional face of C, specifically a half-line \{ \lambda r \mid \lambda \geq 0 \} where r \neq 0 and r \in C, such that the ray cannot be expressed as a nontrivial conic combination of other points in C. Equivalently, if r = y + z with y, z \in C, then y and z must both be nonnegative scalar multiples of r. This irreducibility ensures that extreme rays form the "edges" of the cone, analogous to extreme points in compact convex sets.^[37] For a pointed convex cone (one containing no nontrivial linear subspace), the cone C is the conic hull of its extreme rays, meaning every element of C can be expressed as a nonnegative linear combination of directions along these rays. If C is closed, this extends to the closed conic hull. This representation serves as a Krein-Milman-type theorem for cones: in finite dimensions, every closed pointed convex cone is the closed conic hull of its (possibly infinitely many) extreme rays.^[39]^[37] In the special case of polyhedral cones, defined as the conic hull of finitely many vectors, the number of extreme rays is finite. These rays correspond to the irreducible generators of the cone and can be enumerated algorithmically, providing a minimal finite description of the cone's structure. For example, the nonnegative orthant \mathbb{R}^n_+ has exactly n extreme rays, aligned with the standard basis vectors.^[11]^[40] When the polyhedral cone is rational—meaning it is generated by rational vectors—a Hilbert basis provides a minimal set of integer points that generate the semigroup C \cap \mathbb{Z}^n additively. This basis consists of the irreducible integer elements in C, including directions from the extreme rays scaled to integer form, and every integer point in the cone is a nonnegative integer combination of these basis elements. Computing a Hilbert basis is central to integer programming over rational cones, with algorithms ensuring minimality for pointed cases.^[41]

Closure, Interior, and Relative Interior

The closure of a convex cone C, denoted \operatorname{cl}(C), is the smallest closed convex cone containing C, obtained by adjoining all limit points of sequences in C.^[37] Since the closure operation preserves convexity and the cone property under limits along rays, \operatorname{cl}(C) always contains C and is itself a convex cone.^[37] The interior of a convex cone C, denoted \operatorname{int}(C), consists of all points x \in C such that there exists \varepsilon > 0 with the open ball B(x, \varepsilon) contained in C. This interior is nonempty if and only if C has full dimension in its linear span, meaning the affine hull of C coincides with the ambient space \mathbb{R}^n. For example, the nonnegative orthant \mathbb{R}^n_+ has nonempty interior \mathbb{R}^n_{++}, reflecting its full-dimensional spanning.^[10] The relative interior of a convex cone C, denoted \operatorname{ri}(C), is the interior of C relative to its affine hull \operatorname{aff}(C), defined as

\operatorname{ri}(C) = \{ x \in C \mid \exists \varepsilon > 0, \, B(x, \varepsilon) \cap \operatorname{aff}(C) \subseteq C \}.

Equivalently, \operatorname{ri}(C) is the interior of the translate C - \operatorname{lin}(C) relative to \operatorname{aff}(C), where \operatorname{lin}(C) = C \cap (-C) is the lineality space. Unlike the absolute interior, the relative interior is always nonempty for any nonempty convex cone C. For a closed convex cone C, it holds that C = \operatorname{cl}(\operatorname{ri}(C)), allowing recovery of the cone from its relative interior via closure. In optimization, barrier cones—defined as the set of recession directions approaching the interior of C—facilitate interior-point methods by ensuring strict feasibility within \operatorname{int}(C) or \operatorname{ri}(C).^[10]

Facets and Minimal Faces

In convex geometry, a facet of a convex cone C is defined as a face of C that has codimension one in the linear span of C, meaning its dimension is \dim(\operatorname{lin}(C)) - 1.^[42] This distinguishes facets as the highest-dimensional proper faces, forming the "boundaries" that tightly constrain the cone's structure. For polyhedral cones, facets play a central role in the half-space (H-)representation, where C = \{ x \in \mathbb{R}^n \mid A x \geq 0 \} for some matrix A with rows a_i, and each facet F_i corresponds to a tight inequality: F_i = \{ x \in C \mid a_i \cdot x = 0 \}.^[42] For polyhedral cones, the minimal face containing a given point x \in C is the intersection of all facets of C that contain x.^[43] In the H-representation, this minimal face consists of all points in C satisfying the equalities corresponding to the inequalities tight at x, effectively reducing the cone's dimensionality to capture the local structure around x. For general faces of C, facets serve as building blocks, as every face arises as an intersection of facets. In pointed polyhedral cones, duality establishes a bijection between the facets of C and the extreme rays of its dual cone C^*, implying that the number of facets of C equals the number of extreme rays of C^*.^[44] This correspondence arises from the polar relationship, where each facet-defining hyperplane of C exposes an extreme ray in C^*, and vice versa, highlighting the symmetric combinatorial structure of polyhedral cones under polarity.

Applications

Partial Orders Induced by Cones

A convex cone C in a real vector space V induces a preorder \leq_C on V defined by x \leq_C y if and only if y - x \in C.^[45] This relation is reflexive, since x - x = 0 \in C for all x \in V, and transitive, since if x \leq_C y and y \leq_C z, then z - x = (z - y) + (y - x) \in C + C \subseteq C by the convexity and conic properties of C.^[45] Moreover, the preorder is compatible with the vector space structure: if x \leq_C y, then x + z \leq_C y + z for any z \in V, and \lambda x \leq_C \lambda y for any \lambda \geq 0.^[45] If C is pointed, meaning C \cap (-C) = \{0\}, then \leq_C is antisymmetric: x \leq_C y and y \leq_C x imply y - x \in C and x - y \in C, so y - x \in C \cap (-C) = \{0\}, hence x = y.^[46] In this case, \leq_C defines a partial order on V.^[46] A classic example is the nonnegative orthant C = \mathbb{R}^n_+ in \mathbb{R}^n, which induces the componentwise partial order x \leq_C y if and only if x_i \leq y_i for all i = 1, \dots, n. This order is pointed and widely used in analysis to model nonnegativity constraints. In economics and finance, cone-induced partial orders provide a foundation for comparing uncertain outcomes without full utility specifications.^[47] Seminal work by Brumelle and Vickson unified various stochastic dominance criteria—such as first- and second-order dominance—by representing them as membership in convex cones generated by utility classes or integral transforms of distributions.^[48] Post-2010 literature has extended these ideas to multivariate settings, where cone orders enable distributionally robust portfolio optimization by defining dominance relations over joint return distributions based on convex cone preferences, improving risk assessment in high-dimensional assets.^[49]

Role in Optimization and Duality

Convex cones play a pivotal role in conic optimization, a framework that generalizes linear and semidefinite programming by incorporating constraints defined over proper convex cones. A standard conic optimization problem seeks to minimize \mathbf{c}^\top \mathbf{x} subject to \mathbf{x} \in C, A\mathbf{x} = \mathbf{b}, where C is a closed convex cone and A is a linear map; the problem is feasible if the relative interior of C intersects the affine set \{ \mathbf{x} \mid A\mathbf{x} = \mathbf{b} \}. This structure ensures the feasible set is convex, enabling efficient interior-point methods for solution.^[10] Duality in conic optimization leverages the dual cone C^* = \{ \mathbf{y} \mid \mathbf{y}^\top \mathbf{z} \geq 0 \ \forall \mathbf{z} \in C \}, which appears in the Lagrangian dual formulation: maximize -\mathbf{b}^\top \boldsymbol{\nu} subject to A^\top \boldsymbol{\nu} + \mathbf{s} = \mathbf{c}, \mathbf{s} \in C^*. Weak duality holds universally, with the dual objective providing a lower bound on the primal value, but strong duality—equality of optimal values—requires conditions like Slater's, which posits the existence of a strictly feasible point in the relative interior of the primal cone constraints. This condition guarantees zero duality gap and attainment of optima, facilitating sensitivity analysis and algorithmic convergence.^[10]^[50] Linear programming exemplifies conic optimization over polyhedral cones, specifically the non-negative orthant \mathbb{R}^n_+, where constraints \mathbf{x} \geq \mathbf{0} translate to \mathbf{x} \in \mathbb{R}^n_+; strong duality holds under feasibility due to the cone's self-duality. Semidefinite programming, conversely, employs the positive semidefinite cone \mathcal{S}^n_+ of symmetric matrices, as in minimizing \operatorname{tr}(C X) subject to \operatorname{tr}(A_i X) = b_i, X \succeq 0, which captures quadratic and spectral constraints efficiently.^[10] Cone constraints broadly generalize linear inequalities, replacing \mathbf{x} \geq \mathbf{0} with membership in arbitrary convex cones to model diverse phenomena like robustness in control systems. In the 2020s, advances in sum-of-squares optimization have utilized spectrahedra—feasible sets defined by linear matrix inequalities over semidefinite cones—to certify polynomial non-negativity, enabling scalable hierarchies for global optimization with improved bounds via moment relaxations.^[10]^[51]

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