arXiv subjects

Tom-Lukas Kriel

Publications and source records attributed to Tom-Lukas Kriel.

4 recordsLinked to original sources

arXiv2018

On the exactness of Lasserre relaxations and pure states over real closed fields

Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq\mathbb R^n$. Its Lasserre relaxation of degree $d$ is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most $d$. It defines a spectrahedron that projects down to a convex semialgebraic set containing $S$. In the best case, the projection equals the convex hull of $S$. We show that this is very often the case for sufficiently high $d$ if $S$ is compact and "bulges outwards" on the boundary of its convex hull. Now let additionally a polynomial objective function $f$ be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree $d$ is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if $S$ is compact and $d$ exceeds some bound that depends on the description of $S$ and certain characteristicae of $f$ like the mutual distance of its global minimizers on $S$.

math.AG↗

arXiv2018

An introduction to matrix convex sets and free spectrahedra

The purpose of this paper is to give a self-contained overview of the theory of matrix convex sets and free spectrahedra. We will give new proofs and generalizations of key theorems. However we will also introduce various new concepts and results as well. Key contributions of this paper are: - A new general Krein-Milman theorem that characterizes the smallest operator tuple defining a compact matrix convex set. - The introduction and a characterization of matrix exposed points. - A (weak) Minkowski theorem in the language of matrix extreme points (with a new proof of the weak Krein-Milman theorem of Webster and Winkler). - Simplified/new proofs of the Gleichstellensatz, Helton and McCulloughs characterization of free spectrahedra as closures of matrix convex "free basic open semialgebraic" sets and a characterization of hermitian irreducible free loci of Helton, Klep and Vol$\check{\text{c}}$i$\check{\text{c}}$.

math.AG↗

arXiv2018

On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. In their seminal work of 2009, Helton and Nie showed under the same conditions that $S$ is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a non-constructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of $S$.

math.AG↗

arXiv2016

A new proof for the existence of degree bounds for Putinar's Positivstellensatz

Putinar's Positivstellensatz is a central theorem in real algebraic geometry. It states the following: If you have a set $S= \{ x \in R^n \ | \ g_1 (x) \geq 0, ... , g_m(x) \geq 0\}$ described by some real polynomials $g_i$, then every real polynomial $f$ that is positive on $S$ can be written as a sum of squares weighted by the $g_i$ and $1$. Consider such an identity $f= \sum_{i=1}^{m} g_i s_i + s_0$. For the applications in polynomial optimization, especially semidefinite programming, the following is important: There exists a bound $N$ for the degrees of the $s_i$ which depends only on the $g_i$, $n$, the degree of $f$, an upper bound for $||f||$ and a lower bound for $\min f(S)$. Two proofs from Prestel and Heß resp. Schweighofer and Nie ([Pr], [He] resp. [Sw], [NS]) for the existence of these degree bounds are known (also for the matrix version of Putinar's Positivstellensatz by Helton and Nie [HN]). Prestel uses valuation and model theory for his approach while Schweighofer gives a constructive solution by using a theorem of Pólya. In this paper we will give a new elementary, short but non-constructive proof.

math.AG↗