Seminars

In science and engineering, we regularly face (constrained) polynomial optimization problems (CPOP). That is the task to minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying nonnegativity of real polynomials – a key problem in real algebraic geometry since the 19th century. Since this is a notoriously hard to solve problem (e.g., various NP-complete problems admit a CPOP formulation), one is interested in certificates that imply nonnegativity and are easier to check than nonnegativity itself. Many applications require solving CPOPs that have a sparse structure, i.e., given the n variables and maximal total degree d, just very few of the possible (n+d) terms appear. In this talk, I will give a brief general introduction to polynomial optimization, recall key developments from the last decade regarding certain certificates with advantages in sparse settings, and then give an overview on some of our recent and ongoing work in this area both with a theoretical and an applied focus.