GloptiNets

This is the code of the paper GloptiNets: Scalable Non-Convex Optimization with Certificates.

Install

This code was executed on Julia 1.8.5. First instantiate the project with ] instantiate. Then you can run the files in the REPL. CUDA compatible GPU are recommended for interpolation.

Code structure

I did not build the documentation, but here is an outline of the code structure to get you started.

.
├── src
|   ├── GloptiNets.jl        # Training loop in the `interpolate` function and certificate computation
|   ├── psdmodels.jl         # Block PSD models, in Fourier or Chebychev basis
|   ├── polynomials.jl       # Same for polynomials
|   ├── besselsampler.jl     # Sampling from the Bessel distribution
|   └── experimental.jl      # Pieces of code which might turn out being useful
├── scripts                  # For running experiments and be used in the REPL
|   ├── xps                  # Launch large scale experiments to produce the figures of the paper 
|   ├── process_xps          # Process the experiments and produce plots in the REPL
|   ├── gen_random_poly.jl   # Generate the random polynomial for the experiments
|   ├── ...                  # Some scripts which might turn out being useful 
├── data
|   ├── hnorm2               # Polynomials with H norm between 1 and 20 for Fig. 3
|   └── vs_tssos             # Polynomials for Figs. 1 and 2
...

You can use the load(::Union{PolyCheby, PolyTrigo}, filename), to load the random polynomial and inspect them. See scripts/tssos_{cheby, trigo}.jl to see how to convert them to DynamicPolynomials.

Scripts

• Optionally remove scripts unrelated to the paper, {cheby, fourier}_speed.jl.

Possible improvements

• If sampling more frequencies from the probability distribution to compute the MoM estimator is necessary, then it would be more interesting to sample the frequencies with samplesprobas_bycat_wgrid instead of samplesprobas_bycat which uses a hash table.
• Once #722 is solved, merge branch julia19 which rm dropdims

Citation

Gaspard Beugnot, Julien Mairal, & Alessandro Rudi (2023). GloptiNets: Scalable Non-Convex Optimization with Certificates. arXiv:2306.14932