Randomized SDP solver with Riemannian Hamiltonian Monte Carlo

The relationship between sampling and optimization has gained increasing interest in recent years. A great number of optimization algorithms that are developed leverage sampling. The goal of this project is to equip the VolEsti library with a Riemannian Hamiltonian Monte Carlo (RHMC) sampler constrained on convex bodies and then use it for the implementation of optimization algorithms. The implemented sampler should return samples from log-concave distributions constrained in polytopes as well as spectrahedra. First of all, the initial implementation will be based on very recent promising research about an RHMC algorithm constrained in polytopes. We will expand on this effort by extending its capabilities to handle Spectrahedra. Afterwards, the above implementation will be used as a subroutine for a Semidefinite Program (SDP) solver. Last but not least, we will also include benchmark tests to compare the efficiency of our code against other state of the art libraries. As a summary, the project's deliverables will consist of the implementation of: an RHMC sampler first constrained on polytopes, then extended on spectrahedra, SDP optimization algorithms that use RHMC and finally testing against other libraries.