jump-dev
Add n-dimensional PowerCone
@lkapelevich and @HFriberg have been emailing me about the upcoming CBF v4, https://cblib.zib.de/doc/format4.pdf, which proposes to add an n-dimensional PowerCone.
We've talked about this somewhat before: https://github.com/jump-dev/MathOptInterface.jl/issues/977
There are two proposed variants (plus their dual cones).
I have no strong feelings on which ones (if any) we should adopt. But I'd like to somewhat reign in having a litany of very similar cones.
Radial
Hypograph
Clarabel
Clarabel has native support for the radial with the restriction that $\sigma=1$ https://clarabel.org/stable/api_cone_types/
Hypatia
Hypatia has both https://jump.dev/Hypatia.jl/stable/api/cones/#Hypatia.Cones.GeneralizedPower and https://jump.dev/Hypatia.jl/stable/api/cones/#Hypatia.Cones.HypoPowerMean
Mosek
Mosek has support for radial: https://docs.mosek.com/latest/capi/alphabetic-functionalities.html#mosek.task.appendprimalpowerconedomain
Exponent vector All implementations require a normalized vector of exponents, $\sum_i \alpha_i = 1$, so the only difference is whether you try to assert on this floating-point relation using code such as in Clarabel:
@assert isapprox(sum(α),one(DefaultFloat), atol=eps()*length(α)/2)
or you simply rescale whatever you get in
α = α ./ sum(α)
The latter option avoids a critical loss in precision already at the modeling stage, as the user can input [1, 1, 1], rather than some [0.3333, 0.3333, 0.3333], thereby letting the solver decide on the precision of the decimal expansion.
Radial or hypograph? The boring fact that the Radial variant is supported in all mentioned solvers, means that there should be strong arguments for choosing something else. But for completeness, I want to mention that I have seen three variants
(x^a * y^b * z^c)^(1/(a+b+c)) > t > 0
(x^a * y^b * z^c)^(1/(a+b+c)) > |t|
(x^a * y^b * z^c)^(1/(a+b+c)) > t
In the former two, you can move the power to the other side (e.g., x^a * y^b * z^c > |t|^(a+b+c)) which is nice cognitive simplification from a modeling perspective, as something like t > x^7 is easily recognized as t * 1^6 > x^7 . In contrast, I have yet to find an example where the negative range of t is natural and useful.
We discussed this on today's developer call. We're happy to add the radial power cone if it is supported by CBF and Mosek.