Power cone

In linear algebra, a power cone is a kind of a convex cone that is particularly important in modeling convex optimization problems.[1] 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 {\displaystyle 0<r<1} . It is defined as:[1]

P n , r , 1 − r := { x ∈ R n :     x 1 ≥ 0 ,     x 2 ≥ 0 ,     x 1 r ⋅ x 2 1 − r ≥ x 3 2 + ⋯ + x n 2 } {\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\}}

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]

References

  1. ^ a b c "MOSEK Modeling Cookbook - the Power Cones".


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