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Positivstellensätze for semirings
In this paper we develop a number of results and notions concerning Positivstellensätze for semirings (preprimes) of commutative unital real...
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Positivstellensätze for polynomial matrices
In this paper we establish some Positivstellensätze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for...
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Archimedean Representation Theorem for modules over a commutative ring
Pólya’s Positivstellensatz and Handelman’s Positivstellensatz are known to be concrete instances of the abstract Archimedean Representation Theorem...
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Archimedean Quadratic Modules and Positivstellensätze
Let Q be a quadratic module of a unital \(*\) -algebra A. The set of bounded elements of A with respect to Q forms a \(*\) -subalgebra which...
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Convergence rates for sums-of-squares hierarchies with correlative sparsity
This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of...
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State polynomials: positivity, optimization and nonlinear Bell inequalities
This paper introduces state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. A state analog of Artin’s...
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Sum-of-squares chordal decomposition of polynomial matrix inequalities
We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the...
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Recent Developments
In this short chapter, a number of important developments and advances are summarized that mostly occurred after the 1989 publication of Einführung...
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Introduction and Preliminaries
In this first chapter we give an introduction and outline of the topics from the book. We also introduce basic notions and results from linear...
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Relative Entropy Methods in Constrained Polynomial and Signomial Optimization
Relative entropy programs belong to the class of convex optimization problems. Within techniques based on the arithmetic-geometric mean inequality,...
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Representation of positive polynomials on a generalized strip and its application to polynomial optimization
We study the representation of nonnegative polynomials in two variables on a certain class of unbounded closed basic semi-algebraic sets (which are...
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Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
A noncommutative (nc ) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra...
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An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space
This textbook provides an introduction to representations of general ∗-algebras by unbounded operators on Hilbert space, a topic that naturally...
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A Generalization of Strassen’s Theorem on Preordered Semirings
Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by...
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Spectrahedral Shadows
In this chapter we introduce the notion of a spectrahedral shadow, and examine these sets thoroughly. They are still feasible sets for semidefinite...
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Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere
We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special...