Back to the power cones: we were excited about this feature because it allows us to implement market impact constraints with exponents that are arbitrary greater than one.
We benchmarked a simple portfolio optimization problem. The goal was to maximize an expected return on a 10 000 instrument portfolio under a market impact constraint. We chose this particular problem because it contains ten thousand power cones in order to implement the exponent part of the market impact constraint.
With Mosek 8.1 we were limited to powers 1, 3/2, 5/3 and 2 because other powers would have taken a gigantic effort to implement. With Mosek 9 all the powers above one are now easily available. As can be seen in the following timings (in seconds), in our case, the power cones are roughly as fast as the implementation using multiple rotated quadratic cones. This is great news as we were expecting a 10x degradation. Some oddities though: power 5/3 seems to be slower in Mosek 9 than in Mosek 8.1 and power 2 seemed faster in power cones than in rotated cones.
| Version | Mosek 8.1 | Mosek 9.0 | Mosek 9.0 | |
| Implementation | Rotated Cones | Rotated Cones | Power Cones | |
| Exponent | 3/2 | 2.41 | 2.37 | 5.23 |
| 5/3 | 7.84 | 11.53 | 6.93 | |
| 2 | 2.95 | 2.93 | 2.65 | |
| 3.14 | N/A | N/A | 10.87 |
In conclusion, we are really happy with Mosek 9. The power cones are going to replace all the rotated cone shenanigans we previously had, they are as fast and a lot more powerful/simpler to use.
