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An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [8] A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone.
The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}} is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a , the vector ax is in C . [ 2 ]
In mathematics, an 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 initiated by Ernest Vinberg and Bertram Kostant .
In mathematics, especially convex analysis, the recession cone of a set is a cone containing all vectors such that recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
In linear algebra, a power cone is a kind of a convex cone that is particularly important in modeling convex optimization problems. [1] [2] It is a generalization of the quadratic cone: the quadratic cone is defined using a quadratic equation (with the power 2), whereas a power cone can be defined using any power, not necessarily 2.
A second-order cone program (SOCP) is a convex optimization problem of the form minimize ...
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets = = ( ()). The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice .