restrictedcone

Restrictedcone is a term used in convex analysis to describe a convex cone formed by restricting a base cone with linear constraints. Given a cone C in a real vector space V and a linear subspace L ⊆ V, the restrictedcone is defined as K = C ∩ L. Equivalently, if L is described by Ax = 0, then K = { x ∈ C : Ax = 0 }. The construction preserves conicity and convexity when the restriction subspace passes through the origin. If C is closed or polyhedral, the restrictedcone is likewise closed or polyhedral.

Properties of a restrictedcone follow from those of C and L. It is convex by construction, and if C is closed (or polyhedral), K is closed (or polyhedral). The dimension of K is typically reduced relative to C, reflecting the imposed linear constraints. Faces and extreme rays of K relate to those of C via intersection: each face of K is the intersection of a face of C with L. If the restricting subspace does not pass through the origin, the intersection need not be a cone, so restrictedcone constructions usually require L to be a linear subspace through the origin.

Examples help illustrate the concept. In R^3, let C be the nonnegative orthant { (x, y, z) : x ≥ 0, y ≥ 0, z ≥ 0 } and let L be the subspace { (x, y, z) : x − y = 0 }. The restrictedcone K = C ∩ L consists of vectors with x = y ≥ 0 and z ≥ 0. This kind of restriction is common in conic programming and optimization.

Applications of restrictedcone include modeling feasible regions in conic optimization with linear constraints, decomposition of problems with multiple conic requirements, and analyses of feasible directions under linear restrictions. See also convex cone, polyhedral cone, and conic programming.