Power cone

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:

$P_{n, r, 1 - r} := {𝐱 \in ℝ^{n} : x_{1} \geq 0, x_{2} \geq 0, x_{1}^{r} \cdot x_{2}^{1 - r} \geq \sqrt{x_{3}^{2} + \dots + x_{n}^{2}}}$

An alternative definition is

$P_{r, 1 - r} := {𝐱_{𝟏}, 𝐱_{𝟐}, 𝐱_{𝟑} : x_{1} \geq 0, x_{2} \geq 0, x_{1}^{r} \cdot x_{2}^{1 - r} \geq | x_{3} |}$

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.

References

1 2 3 "MOSEK Modeling Cookbook - the Power Cones".
↑ Nesterov, Yurii (2006). Towards nonsymmetric conic optimization.

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[:0-1] 1 2 3 "MOSEK Modeling Cookbook - the Power Cones".

[2] Nesterov, Yurii (2006). Towards nonsymmetric conic optimization.