Cone (convex analysis)

A cone is a set closed under multiplication by nonnegative scalars. In convex analysis, it is often also required to be convex, so it is closed under addition.

Convex set: A set containing every line segment between any two of its points. A convex cone adds nonnegative scaling to this defining closure property.
Conic combination: A finite sum of vectors multiplied by nonnegative coefficients. Convex cone closure is equivalent to containing every conic combination of its members.
Conic optimization: Optimization in which constraints require a vector-valued expression to belong to a convex cone. Cone membership supplies a common form for many convex optimization constraints.
Subspace: A subset of a vector space closed under addition and multiplication by every scalar. Unlike a cone, a subspace must also contain negatives of all its vectors.
Farkas' lemma: A theorem giving an alternative between solvability of a linear system and a nonnegative certificate of infeasibility. It turns cone membership into a precise alternative between solutions and separating inequalities.
Nonnegative scalar: A real number greater than or equal to zero. Scaling by these numbers is the defining closure operation for a cone.
Cone generated by a set: The set of all conic combinations of members of a given set. It constructs the smallest convex cone containing specified generators.
Second-order cone programming: Convex optimization with affine constraints in second-order cones and a linear objective. It uses the Lorentz cone to express norm bounds and related convex constraints.
Cone of recession: The directions along which a convex set can be translated indefinitely while remaining inside it. It is a cone associated with a convex set, not necessarily the set itself.
Supporting hyperplane theorem: A theorem guaranteeing a hyperplane that supports a convex set at a boundary point under suitable conditions. Supporting hyperplanes expose faces and boundaries of convex cones.

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PolyhedronCompared with: A polyhedral cone is a special unbounded polyhedron with its apex at the origin.