C is a cone if x ∈ C {\displaystyle x\in C} implies s x ∈ C {\displaystyle sx\in C} for every positive scalar s. A cone need not be convex, or even...

Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. The dual cone C* of a subset C in a linear space X...

a pyramid. Depending on the context, "cone" may also mean specifically a convex cone or a projective cone. Cones can also be generalized to higher dimensions...

invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was...

tangent cone, including the adjacent cone, Bouligand's contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but...

convex program can be presented in a conic form, which means minimizing a linear objective over the intersection of an affine plane and a convex cone:: 5...

cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp...

is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region...

subsets, convex cones, and convex sets are generalizations of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set...

bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence...

A second-order cone program (SOCP) is a convex optimization problem of the form minimize   f T x   {\displaystyle \ f^{T}x\ } subject to ‖ A i x + b i...

Thus we can take, for example, y1 = 1, y2 = −2. Consider the closed convex cone C ( A ) {\displaystyle C(\mathbf {A} )} spanned by the columns of A;...

then the closed cone of curves is the closure of the positive cone. Otherwise, the closed cone of curves is the closed convex cone spanned by all elements...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

prism), and a polyhedral cone (infinite cone) defined by three or more half-spaces passing through a common point. A convex polytope may be defined in...

of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone. The...

is a finite-rank free abelian group. A strongly convex rational polyhedral cone in N is a convex cone (of the real vector space of N) with apex at the...

the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar...

of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar...

hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone). For several parameters,...

semidefinite matrices forms a convex cone in Rn × n, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace. Spectrahedra...

In mathematics, especially convex analysis, the recession cone of a set A {\displaystyle A} is a cone containing all vectors such that A {\displaystyle...

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group...

hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone). For several parameters,...

cone is a kind of a convex cone that is particularly important in modeling convex optimization problems. It is a generalization of the quadratic cone:...

matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite...

positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone. Some authors use more general definitions...

realization of an abstract polytope is a convex cone of infinite dimension.: 127 : 229–230  The realization cone of the abstract apeirogon has uncountably...

b+L is an affine plane), and K is a closed pointed convex cone with a nonempty interior. Every convex program can be converted to the conic form. To use...

Thus it links operator algebraic and convex order theoretic views on state spaces of physical systems. A convex cone C {\displaystyle C} is called regular...