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Über dieses Buch

This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.



Chapter 1. Selected Results from Algebra and Mathematical Optimization

Positive semidefinite matrices will be used extensively throughout the book. Therefore we fix notation here and present some basic properties needed later on.

Sabine Burgdorf, Igor Klep, Janez Povh

Chapter 2. Detecting Sums of Hermitian Squares

The central question of this chapter is how to find out whether a given nc polynomial is a sum of hermitian squares (SOHS). We rely on Sect. 1.3, where we explained basic relations between SOHS polynomials and positive semidefinite Gram matrices. In this chapter we will enclose these results into the Gram matrix method and refine it with the Newton chip method.

Sabine Burgdorf, Igor Klep, Janez Povh

Chapter 3. Cyclic Equivalence to Sums of Hermitian Squares

When we move focus from positive semidefinite non-commutative polynomials to trace-positive non-commutative polynomials we naturally meet cyclic equivalence to hermitian squares, see Definitions 1.57 and 1.60 In this chapter we will consider the question whether an nc polynomial is cyclically equivalent to SOHS, i.e., whether it is a member of the cone $$\varTheta ^{2}$$ , which is a sufficient condition for trace-positivity. A special attention will be given to algorithmic aspects of detecting members in $$\varTheta ^{2}$$ . We present a tracial version of the Gram matrix method based on the tracial version of the Newton chip method which by using semidefinite programming efficiently answers the question if a given nc polynomial is or is not cyclically equivalent to a sum of hermitian squares.

Sabine Burgdorf, Igor Klep, Janez Povh

Chapter 4. Eigenvalue Optimization of Polynomials in Non-commuting Variables

In Sect. 1.6 we introduced a natural notion of positivity that corresponds exactly to nc polynomials that are SOHS. Recall that an nc polynomial is positive semidefinitenc polynomialpositive semidefinite if it yields a positive semidefinite matrix when we replace the letters (variables) in the polynomial by symmetric matrices of the same order. Helton’s Theorem 1.30 implies that positive semidefinite polynomials are exactly the SOHS polynomials, the set of which we denoted by Σ2.

Sabine Burgdorf, Igor Klep, Janez Povh

Chapter 5. Trace Optimization of Polynomials in Non-commuting Variables

In Chap. 3 trace-positivity together with the question how to detect it was explored in details. Due to hardness of the decision problem “Is a given nc polynomial f trace-positive?” we proposed a relaxation of the problem, i.e., we are asking if f is cyclically equivalent to SOHS. The tracial Gram matrix method based on the tracial Newton polytope was proposed (see Sects. 3.3 and 3.4) to efficiently detect such polynomials.

Sabine Burgdorf, Igor Klep, Janez Povh


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