Dr. Benoît Legat
Research – Effective Polynomial Optimization leveraging the numerical robustness of the Macaulay framework and the convex computational features of the Sum-of-Squares / Moment hierarchy.
Sum-of-Squares is an effective method to find polynomial functions satisfying given properties by via semidefinite programming. One of its main drawbacks for polynomial optimization is that we can currently only provide the optimal solutions at a high level of the hierarchy. Another approach is to solve the polynomial system formed by the Karush–Kuhn–Tucker conditions using numerical linear algebra via the Macaulay framework developed in this ERC. The drawback of this approach is that the system contains all first-order critical point, including spurious ones, i.e., local minimizers that are not global minimizers but also local maximizers and saddle points. This large number of solutions will slow down the numerical solution process. My aim is to combine both Sum-of-Squares and Macaulay approaches into an effective algorithm for polynomial optimization.