Power cone

Short description: Concept in linear algebra

Not to be confused with Cone of power.

In linear algebra, a power 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: 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.

Definition

The n-dimensional power cone is parameterized by a real number

$0<r<1$

. It is defined as:

$P_{n,r,1-r}:=\{\mathbf{𝐱}\in {\mathbb{ℝ}}^n:x_1\ge 0,x_2\ge 0,x_1^r\cdot x_2^{1-r}\ge \sqrt{x_3^2+\cdots +x_n^2}\}$

An alternative definition is

$P_{r,1-r}:=\{{\mathbf{x}}_{\mathbf{𝟏}},{\mathbf{x}}_{\mathbf{𝟐}},{\mathbf{x}}_{\mathbf{𝟑}}:x_1\ge 0,x_2\ge 0,x_1^r\cdot x_2^{1-r}\ge |x_3|\}$

Applications

The main application of the power cone is in constraints of convex optimization programs. There are many problems that can be described as minimizing a convex function over a power cone.