Exponential lower bounds on spectrahedral representations of hyperbolicity cones

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables contains $(n/d)^{\Omega(d)}$ pairwise distant cones in a certain metric, and therefore that any semidefinite representation of such cones must have dimension at least $(n/d)^{\Omega(d)}$ (even if a small approximation is allowed). The proof contains several ingredients of independent interest, including the identification of a large subspace in which the elementary symmetric polynomials lie in the relative interior of the set of hyperbolic polynomials, and quantitative versions of several basic facts about real rooted polynomials.Comment: Fixed a mistake in the proof of Lemma 6. The statement is unchanged except for constant factors, and the main theorem is unaffected. Wrote a slightly stronger statement for the main theorem, emphasizing approximate representations (the proof is the same). Added one figur

Spectrahedral relaxations of hyperbolicity cones

Let $p$ be a real zero polynomial in $n$ variables. Then $p$ defines a rigidly convex set $C(p)$. We construct a linear matrix inequality of size $n+1$ in the same $n$ variables that depends only on the cubic part of $p$ and defines a spectrahedron $S(p)$ containing $C(p)$. The proof of the containment uses the characterization of real zero polynomials in two variables by Helton and Vinnikov. We exhibit many cases where $C(p)=S(p)$. In terms of optimization theory, we introduce a small semidefinite relaxation of a potentially huge hyperbolic program. If the hyperbolic program is a linear program, we introduce even a finitely convergent hierachy of semidefinite relaxations. With some extra work, we discuss the homogeneous setup where real zero polynomials correspond to homogeneous polynomials and rigidly convex sets correspond to hyperbolicity cones. The main aim of our construction is to attack the generalized Lax conjecture saying that $C(p)$ is always a spectrahedron. To this end, we conjecture that real zero polynomials in fixed degree can be "amalgamated" and show it in three special cases with three completely different proofs. We show that this conjecture would imply the following partial result towards the generalized Lax conjecture: Given finitely many planes in $\mathbb R^n$, there is a spectrahedron containing $C(p)$ that coincides with $C(p)$ on each of these planes. This uses again the result of Helton and Vinnikov.Comment: very preliminary draft, not intended for publicatio

Spectral linear matrix inequalities

We prove, under a certain representation theoretic assumption, that the set of real symmetric matrices, whose eigenvalues satisfy a linear matrix inequality, is itself a spectrahedron. The main application is that derivative relaxations of the positive semidefinite cone are spectrahedra. From this we further deduce statements on their Wronskians. These imply that Newton's inequalities, as well as a strengthening of the correlation inequalities for hyperbolic polynomials, can be expressed as sums of squares

Real Algebraic Geometry With A View Toward Systems Control and Free Positivity

New interactions between real algebraic geometry, convex optimization and free non-commutative geometry have recently emerged, and have been the subject of numerous international meetings. The aim of the workshop was to bring together experts, as well as young researchers, to investigate current key questions at the interface of these fields, and to explore emerging interdisciplinary applications

Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability

Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies

Hyperbolic Secant Varieties of M-Curves

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces.Comment: 32 pages, 2 figures; Comments welcome

Real Algebraic Geometry With a View Toward Moment Problems and Optimization

Continuing the tradition initiated in MFO workshop held in 2014, the aim of this workshop was to foster the interaction between real algebraic geometry, operator theory, optimization, and algorithms for systems control. A particular emphasis was given to moment problems through an interesting dialogue between researchers working on these problems in finite and infinite dimensional settings, from which emerged new challenges and interdisciplinary applications

Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the positive semidefinite matrix cone and the second order cone as important practical examples. In this paper, we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual. We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. We describe transitive subsets of the automorphism groups of the cones and their duals, and important properties of the composition of log-det barrier functions with the automorphisms in this set. Next, we consider extensions to linear slices of the positive semidefinite cone, i.e., intersection of the positive semidefinite cone with a linear subspace, and review conditions that make the cone homogeneous. In the third part of the paper we give a high-level overview of the classical algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation. We conclude by discussing the role of homogeneous cone structure in primal-dual symmetric interior-point methods.Comment: 59 pages, 10 figures, to appear in Acta Numeric