Optimization of Polynomials in Non-Commuting Variables

This bookpresents recent results on positivity and optimization of polynomials innon-commuting variables.

Researchers in non-commutative algebraic geometry,control theory, system engineering, optimization, quantum physics andinformation science will find the unified notation and mixture of algebraicgeometry and mathematical programming useful.

Theoretical results are matchedwith algorithmic considerations; several examples and information on how to useNCSOStools open source package to obtain the results provided.

Results arepresented on detecting the eigenvalue and trace positivity of polynomials innon-commuting variables using Newton chip method and Newton cyclic chip method,relaxations for constrained and unconstrained optimization problems, semidefiniteprogramming formulations of the relaxations and finite convergence of thehierarchies of these relaxations, and the practical efficiency of algorithms.