Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 166)

Abstract

One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called “dimension-free”. Analyzing dimension-free problems has led to the development recently of a non-commutative (nc) real algebraic geometry (RAG) which, when combined with convexity, produces dimension-free Semidefinite Programming. This article surveys what is known about convexity in the non-commutative setting and nc SDP and includes a brief survey of nc RAG. Typically, the qualitative properties of the non-commutative case are much cleaner than those of their scalar counterparts – variables in \({\mathbb{R}}^{g}\). Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • J. William Helton
    • 1
  • Igor Klep
    • 2
    • 3
  • Scott McCullough
    • 4
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  2. 2.Univerza v Ljubljani, Fakulteta za matematiko in fizikoLjubljanaSlovenija
  3. 3.Univerza v Mariboru, Fakulteta za naravoslovje in matematikoMariborSlovenija
  4. 4.Department of MathematicsUniversity of FloridaGainesvilleUSA

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