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.[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

0<r<1

. It is defined as:[1]

> Pn,r,1−r:={𝐱∈ℝn:x1≥0,x2≥0,x1r⋅x21−r≥x32+⋯+xn2}

An alternative definition is

> Pr,1−r:={x𝟏,x𝟐,x𝟑:x1≥0,x2≥0,x1r⋅x21−r≥|x3|}

## 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. ↑ 1.0 1.1 1.2 "MOSEK Modeling Cookbook - the Power Cones". https://docs.mosek.com/modeling-cookbook/powo.html.
2. ↑ Nesterov, Yurii (2006). Towards nonsymmetric conic optimization.

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