Power cone

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.^[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.

Definition

The n-dimensional power cone is parameterized by a real number ${\displaystyle 0<r<1}$. It is defined as:^[1]

${\displaystyle P_{n,r,1-r}:=\left\{\mathbf {x} \in \mathbb {R} ^{n}:~~x_{1}\geq 0,~~x_{2}\geq 0,~~x_{1}^{r}\cdot x_{2}^{1-r}\geq {\sqrt {x_{3}^{2}+\cdots +x_{n}^{2}}}\right\}}$

An alternative definition is

${\displaystyle P_{r,1-r}:=\left\{\mathbf {x_{1},x_{2},x_{3}}$ :~~x_{1}\geq 0,~~x_{2}\geq 0,~~x_{1}^{r}\cdot x_{2}^{1-r}\geq |x_{3}|\right\}}

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.^[1]