A spectrahedron is an affine-linear slice of the cone of positive
semidefinite matrices. Spectrahedra are the domains of semidefinite
programs, just as polyhedra are those of linear programs. The
boundary of a spectrahedron is contained in an algebraic
hypersurface satisfying a reality condition called
hyperbolicity. However, it remains mysterious whether
hyperbolicity suffices to characterize spectrahedra. The exact
relation is the content of the Generalized Lax Conjecture,
and making progress on this conjecture is one of my main goals. There are
surprising connections to sums of squares of polynomials
that I am currently investigating.
Positive polynomials and sums of squares
Connections
between positive polynomials and sums of squares pervade real
algebraic geometry. When a polynomial is expressed as a sum of
squares, its non-negativity is obvious, and testing for such an
expression can be carried out efficiently in a semidefinite
program. Positivity on semi-algebraic sets (i.e. under additional
polynomial inequalities) is expressed by weighted sums of
squares. I work on the existence and complexity of such
representations, especially in the context of the moment problem in
functional
analysis and in
relation with convexity and combinatorics.
Semidefinite representations of convex sets
Many
convex sets that are not spectrahedra can still be expressed as
projected spectrahedra. Such lifted representations are of
great interest in convex optimization. A general machinery for
approximating the convex hull of a semi-algebraic set by a sequence
of projected spectrahedra is the Lasserre
Relaxation, which works by expressing supporting hyperplanes
as weighted sums of squares. A conjecture due to Helton and Nie states that every convex
semi-algebraic set is a projected spectrahedron. A further goal is
to prove lower bounds on the complexity of semidefinite representations.
DFG Project Convexity in Real Algebraic Geometry
Project Abstract. In this project, problems from convexity and from convex and polynomial optimisation are studied with specific methods from real algebraic geometry. This concerns fundamental questions arising from applications in optimisation (in particular semidefinite programming and positive polynomials), but also the classical geometry of curves and surfaces and their higher dimensional analogues. Two aspects that have received a lot of attention in recent years concern the representation of real polynomials by determinants and the description of convex sets by linear matrix inequalities.
