Global optimization of polynomials using gradient tentacles and sums of squares

In this work, the combine the theory of generalized critical values with the theory of iterated rings of bounded elements (real holomorphy rings). We consider the problem of computing the global infimum of a real polynomial in several variables. Every global minimizer lies on the gradient variety. If the polynomial attains a minimum, it is therefore equivalent to look for the greatest lower bound on its gradient variety. Nie, Demmel and Sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds. Based on these certificates, they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently. We deal here with the more general case when the polynomial is bounded from belo w but does not necessarily attain a minimum. In this case, the method of Nie, Demmel and Sturmfels might yield completely wrong results. In order to overcome this problem, we replace the gradient variety by larger semialgebraic sets which we call gradient tentacles. It now gets substantially harder to prove the existence of the necessary sums of squares certificates.Comment: 22 page

On the complexity of Putinar's Positivstellensatz

We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints

A criterion for membership in archimedean semirings

We prove an extension of the classical Real Representation Theorem (going back to Krivine, Stone, Kadison, Dubois and Becker and often called Kadison-Dubois Theorem). It is a criterion for membership in subsemirings (sometimes called preprimes) of a commutative ring. Whereas the classical criterion is only applicable for functions which are positive on the representation space, the new criterion can under certain arithmetic conditions be applied also to functions which are only nonnegative. Only in the case of preorders (i.e., semirings containing all squares), our result follows easily from recent work of Scheiderer, Kuhlmann, Marshall and Schwartz. Our proof does not use (and therefore shows) the classical criterion. We illustrate the usefulness of the new criterion by deriving a theorem of Handelman from it saying inter alia the following: If an odd power of a real polynomial in several variables has only nonnegative coefficients, then so do all sufficiently high powers.Comment: 23 pages. See also: http://www.mathe.uni-konstanz.de/homepages/schweigh

Exposed faces of semidefinitely representable sets

A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie

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