Two results on the size of spectrahedral descriptions

A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron we are interested in the question of the smallest possible size $r$ of the matrices in the description by linear matrix inequalities. We show that for the $n$-dimensional unit ball $r$ is at least $\frac{n}{2}$. If $n=2^k+1$, then we actually have $r=n$. The same holds true for any compact convex set in $\mathbb{R}^n$ defined by a quadratic polynomial. Furthermore, we show that for a convex region in $\mathbb{R}^3$ whose algebraic boundary is smooth and defined by a cubic polynomial we have that $r$ is at least five. More precisely, we show that if $A,B,C$ are real symmetric matrices such that $f(x,y,z)=\det(I+A x+B y+C z)$ is a cubic polynomial, the surface in complex projective three-space with affine equation $f(x,y,z)=0$ is singular.Comment: 10 pages, 2 figures, minor mistakes correcte

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

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

Calculus of unbounded spectrahedral shadows and their polyhedral approximation

The present thesis deals with the polyhedral approximation and calculus of spectrahedral shadows that are not necessarily bounded. These sets are the images of the feasible regions of semidefinite programs under linear transformations. Spectrahedral shadows contain polyhedral sets as a proper subclass. Therefore, the method of polyhedral approximation is a useful device to approximately describe them using members of the same class with a simpler structure. In the first part we develop a calculus for spectrahedral shadows. Besides showing their closedness under numerous set operations, we derive explicit descriptions of the resulting sets as spectrahedral shadows. Special attention is paid to operations that result in unbounded sets, such as the polar cone, conical hull and recession cone. The second part is dedicated to the approximation of compact spectrahedral shadows with respect to the Hausdorff distance. We present two algorithms for the computation of polyhedral approximations of such sets. Convergence as well as correctness of both algorithms are proved. As a supplementary tool we also present an algorithm that generates points from the relative interior of a spectrahedral shadow and computes its affine hull. Finally, we investigate the limits of polyhedral approximation in the Hausdorff distance in general and, extending known results, characterize the sets that admit such approximations. In the last part we develop concepts and tools for the approximation of spectrahedral shadows that are compatible with unboundedness. We present two notions of polyhedral approximation and show that sequences of approximations converge to the true set if the approximation errors diminish. In combination with algorithms for their computation we develop an algorithm for the polyhedral approximation of recession cones of spectrahedral shadows. Finiteness and correctness of all algorithms are proved and properties of the approximation concepts are investigated