A way to approximate the volume of semi algebraic sets using SDPs
On Wednesday 30.01.2020 as part of a reading group I presented the work of: D. Henrion, J. B. Lasserre and C. Savorgnan on: Approximate Volume and Integration for Basic Semi-algebraic Sets. [ https://doi.org/10.1137/080730287 ] I wish to share some interesting bits of knowledge I came across. It is not my intention to fully and rigorously reproduce the contents of the above mentioned paper. Hence, if you are interested in the topic mentioned then please consult the paper. I was suggested this paper by Monique because it shows two things. The first being that finding the volume of a compact basic semi-algebraic set K that is Archemedean (on the polynomials g_i defining K), can be recast as a minimisation problem over measures (see Theorem 3.1). The second thing that is shown is that this problem can be approximated by a hierarchy of SDP approximations that return the moments of a minimising measure. Both of these topics fall well within the scope of POEMA. Clearly the secon