LowRankSOS.jl

Low-rank nonconvex formulation for finding sum-of-square representation of a given quadratic form over a real projective variety. Current examples include rational normal scrolls (scroll), plane cubic curves (cubic), and Veronese varieties (Veronese). See more details in the arXiv preprint.

Run Experiments

Experiments can be run through the run_experiment.jl script under example/, with the following options:

• [example name] which can be scroll, cubic, Veronese, small (default, running all of the previous examples), scroll-large, cubic-large, and large (running both the scroll-large and cubic-large instances); on the smaller instances comparison will be done against Hypatia.jl, Clarabel.jl, SCS, and CSDP SDP solvers, while on the larger instances only CSDP will be included in the output file as comparison.
• [number of repetition] which is the number that we repeat the experiment runs for each setting, default to $100$. Please set it to at least $2$ to produce the csv file output. When this is set to $1$, the script will execute different low-rank methods for demonstration without producing any csv file output!
• [output file name] which is name for the output file and default to result_[example name].csv if omitted. For example, the output file name will be result_large.csv if this argument is omitted and the [example name] is set to be large.

Before official release, the current recommended practice of running the experiments from command line interface is the following, assuming the working directory contains LowRankSOS.jl.

julia --project=LowRankSOS.jl/. LowRankSOS.jl/example/run_experiment.jl [example name] [number of repetition] [output file name]

For comparison against any generic SDP solver, it is recommended to explicitly specify the number of threads for a fair comparison, which can be done by the following command.

export OMP_NUM_THREADS=[number of threads] && export OMP_THREAD_LIMIT=[number of threads] && julia --project=LowRankSOS.jl/. LowRankSOS.jl/example/run_experiment.jl [example name] [number of repetition]