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Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

In: Handbook on Semidefinite, Conic and Polynomial Optimization

Author

Listed:
  • J. William Helton

    (University of California at San Diego)

  • Igor Klep

    (Univerza v Ljubljani, Fakulteta za matematiko in fiziko
    Univerza v Mariboru, Fakulteta za naravoslovje in matematiko)

  • Scott McCullough

    (University of Florida)

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.

Suggested Citation

  • J. William Helton & Igor Klep & Scott McCullough, 2012. "Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 377-405, Springer.
    • Handle: RePEc:spr:isochp:978-1-4614-0769-0_13
    DOI: 10.1007/978-1-4614-0769-0_13
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