Power cone explained

Power cone should not 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] 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

0 <r< 1 . It is defined as:

P _{n,} : = \left\{ x \in R ^{n:~~x} _{1 \geq} 0 ,~~ x _{2\geq} 0 ,~~ x ^{r ⋅} _{1} x ^{1- r} _{2} \geq \sqrt{x ^{2} _{3} + … + x ^{2 } \right\}} _{n}

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.

Notes and References

1. Web site: MOSEK Modeling Cookbook - the Power Cones .