2006/12/04 by J. William Helton, Helton, John William, Mihai Putinar +1 · 4 citations
Computer Science · Engineering · Mathematics · #11E25 #12D15 #47B15 #65K10 #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.math/0612103
openalex publication_date 2006/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free *-algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.