SPECTRA: solving exactly linear matrix inequalities

par Didier Henrion (LAAS-CNRS, Univ. Toulouse, France and Czech Tech. Univ. Prague, Czech Republic)

jeudi 15 déc. 2016, 10:30 → 11:30 Europe/Paris

Salle XR203 (Bâtiment XLIM)

The set of real points such that a symmetric pencil is positive semidefinite is a convex semi-algebraic set called spectrahedron, described by a linear matrix inequality (LMI). We describe our Maple package SPECTRA that computes an exact algebraic representation of at least one point in a given spectrahedron, or decides that it is empty, up to genericity assumptions on the rational input matrices describing the pencil. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, which is polynomial in the number of variables when the size of the pencil is fixed. Our experiments show significant improvements with respect to state-of-the-art computer algebra algorithms. This is joint work with Simone Naldi and Mohab Safey El Din.