Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

  • J. William Helton
  • Igor Klep
  • Scott McCullough
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 166)


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.


  1. 1.
    Arveson, W.: Subalgebras of C  ∗ -algebras. Acta Mathematica 123, 141–224 (1969)CrossRefGoogle Scholar
  2. 2.
    Arveson, W.: Subalgebras of C  ∗ -algebras. II. Acta Mathematica 128(3-4), 271–308 (1972)CrossRefGoogle Scholar
  3. 3.
    Arveson, W.: The noncommutative Choquet boundary. Journal of the American Mathematical Society 21(4), 1065–1084 (2008)CrossRefGoogle Scholar
  4. 4.
    Bochnack, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics, Vol. 36. Springer–Verlag, Berlin (1998)Google Scholar
  5. 5.
    Ball, J.A., Groenewald, G., Malakorn, T.: Bounded real lemma for structured noncommutative multidimensional linear systems and robust control. Multidimensional Systems And Signal Processing 17(2-3), 119–150 (2006)CrossRefGoogle Scholar
  6. 6.
    Burgdorf, S., Klep, I.: The truncated tracial moment problem. To appear in the Journal of Operator Theory, Scholar
  7. 7.
    Blecher, D.P., Le Merdy, C.: Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs, vol. 30. The Clarendon Press Oxford University Press, Oxford (2004)Google Scholar
  8. 8.
    Bessis, D., Moussa, P., Villani, M.: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. Journal of Mathematical Physics 16(11), 2318–2325 (1975)CrossRefGoogle Scholar
  9. 9.
    Ben-Tal A., Nemirovski, A.: On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty. SIAM Journal on Optimization 12(3), 811–833 (2002)CrossRefGoogle Scholar
  10. 10.
    Camino, J.F., Helton, J.W., Skelton, R.E., Ye, J.: Matrix inequalities: a symbolic procedure to determine convexity automatically. Integral Equation and Operator Theory 46(4), 399–454 (2003)CrossRefGoogle Scholar
  11. 11.
    Cafuta, K., Klep, I., Povh, J.: NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials. Optimization methods and Software 26(3), 363–380 (2011)CrossRefGoogle Scholar
  12. 12.
    Connes, A.: Classification of injective factors. Cases { II}1, { II}, { III}λ, λ≠1. Annals of Mathematics 104, 73–115 (1976)Google Scholar
  13. 13.
    Effros, E.G., Winkler, S.: Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems. Journal of Functional Analysis 144(1), 117–152 (1997)CrossRefGoogle Scholar
  14. 14.
    Hay, D.M., Helton, J.W., Lim, A., McCullough, S.: Non-commutative partial matrix convexity. Indiana University Mathematics Journal 57(6), 2815–2842 (2008)CrossRefGoogle Scholar
  15. 15.
    Helton, J.W.: ‘Positive’ noncommutative polynomials are sums of squares. Annals Of Mathematics 156(2), 675–694 (2002)CrossRefGoogle Scholar
  16. 16.
    Helton, J.W., de Oliveira, M.C., Stankus, M., Miller, R.L.: NCAlgebra, 2010 release edition. Available from (2010)
  17. 17.
    Helton, J.W., Klep, I., McCullough, S.: Analytic mappings between noncommutative pencil balls. Journal of Mathematical Analysis and Applications 376(2), 407–428 (2011)CrossRefGoogle Scholar
  18. 18.
    Helton, J.W., Klep, I., McCullough, S.: The matricial relaxation of a linear matrix inequality. Preprint, available from Scholar
  19. 19.
    Helton, J.W., Klep, I., McCullough, S.: Proper analytic free maps. Journal of Functional Analysis 260(5), 1476–1490 (2011)CrossRefGoogle Scholar
  20. 20.
    Helton, J.W., Klep, I., McCullough, S., Slinglend, N.: Noncommutative ball maps. Journal of Functional Analysis 257(1), 47–87 (2009)CrossRefGoogle Scholar
  21. 21.
    Helton, J.W., McCullough, S.: Every free basic convex semi-algebraic set has an LMI representation. Preprint, available from Scholar
  22. 22.
    Helton, J.W., McCullough, S.: Convex noncommutative polynomials have degree two or less. SIAM Journal on Matrix Analysis and Applications 25(4), 1124–1139 (2003)CrossRefGoogle Scholar
  23. 23.
    Helton, J.W., McCullough, S.: A positivstellensatz for non-commutative polynomials. Transactions of The American Mathematical Society 356(9), 3721–3737 (2004)CrossRefGoogle Scholar
  24. 24.
    Helton, J.W., McCullough, S.A., Putinar, M.: A non-commutative positivstellensatz on isometries. Journal Für Die Reine Und Angewandte Mathematik 568, 71–80 (2004)CrossRefGoogle Scholar
  25. 25.
    Helton, J.W., McCullough, S., Putinar, M.: Non-negative hereditary polynomials in a free *-algebra. Mathematische Zeitschrift 250(3), 515–522 (2005)CrossRefGoogle Scholar
  26. 26.
    Helton, J.W., McCullough, S., Putinar, M.: Strong majorization in a free *-algebra. Mathematische Zeitschrift 255(3), 579–596 (2007)CrossRefGoogle Scholar
  27. 27.
    Helton, J.W., McCullough, S.A., Vinnikov, V.: Noncommutative convexity arises from linear matrix inequalities. Journal Of Functional Analysis 240(1), 105–191 (2006)CrossRefGoogle Scholar
  28. 28.
    Helton, J.W., Putinar, M.: Positive polynomials in scalar and matrix variables, the spectral theorem, and optimization. In: Bakonyi, M., Gheondea, A., Putinar, M., Rovnyak, J. (eds.) Operator Theory, Structured Matrices, and Dilations. A Volume Dedicated to the Memory of Tiberiu Constantinescu, pp. 229–306. Theta, Bucharest (2007)Google Scholar
  29. 29.
    Helton, J.W., Vinnikov, V.: Linear matrix inequality representation of sets. Communications On Pure And Applied Mathematics 60(5), 654–674 (2007)CrossRefGoogle Scholar
  30. 30.
    Klep, I., Schweighofer, M.: A nichtnegativstellensatz for polynomials in noncommuting variables. Israel Journal of Mathematics 161(1), 17–27 (2007)CrossRefGoogle Scholar
  31. 31.
    Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of Hermitian squares. Advances in Mathematics 217, 1816–1837 (2008)CrossRefGoogle Scholar
  32. 32.
    Klep, I., Schweighofer, M.: Sums of Hermitian squares and the BMV conjecture. Journal of Statistical Physics 133(4), 739–760 (2008)CrossRefGoogle Scholar
  33. 33.
    Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting. Linear Algebra and its Applications 430(4), 869–889 (2009)CrossRefGoogle Scholar
  34. 34.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11(3), 796–817 (2001)CrossRefGoogle Scholar
  35. 35.
    Lewis, A.S., Parrilo, P.A., Ramana, M.V.: The Lax conjecture is true. roceedings of the American Mathematical Society 133(9), 2495–2499 (2005)Google Scholar
  36. 36.
    McCullough, S.: Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra and its Applications 326(1-3), 193–203 (2001)CrossRefGoogle Scholar
  37. 37.
    Navascués, M., Pironio, S., Acín, A.: SDP relaxations for non-commutative polynomial optimization. this volume.Google Scholar
  38. 38.
    Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology (2000)Google Scholar
  39. 39.
    Paulsen, V.: Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)Google Scholar
  40. 40.
    Prestel, A., Delzell, C.N.: Positive polynomials. From Hilbert’s 17th problem to real algebra. Springer Monographs in Mathematics. Springer, Berlin (2001)Google Scholar
  41. 41.
    Pisier, G.: Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294. Cambridge University Press, Cambridge (2003)Google Scholar
  42. 42.
    Pironio, S., Navascués, M., Acín, A.: Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM Journal on Optimization 20(5), 2157–2180 (2010)CrossRefGoogle Scholar
  43. 43.
    Popescu, G.: Noncommutative transforms and free pluriharmonic functions. Advances in Mathematics 220(3), 831–893 (2009)CrossRefGoogle Scholar
  44. 44.
    Popescu, G.: Free holomorphic functions on the unit ball of B(H)n. II. Journal of Functional Analysis 258(5), 1513–1578 (2010)CrossRefGoogle Scholar
  45. 45.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42(3), 969–984 (1993)CrossRefGoogle Scholar
  46. 46.
    Schmüdgen, K.: A strict Positivstellensatz for the Weyl algebra. Math. Ann. 331(4), 779–794 (2005)CrossRefGoogle Scholar
  47. 47.
    Schmüdgen, K., Savchuk, Y.: Unbounded induced representations of ∗ -algebras. Preprint, Scholar
  48. 48.
    Skelton, R.E., Iwasaki, T., Grigoriadis, K.M.: A unified algebraic approach to linear control design. The Taylor & Francis Systems and Control Book Series. Taylor & Francis Ltd., London (1998)Google Scholar
  49. 49.
    Voiculescu, D.: Free analysis questions II: The Grassmannian completion and the series expansions at the origin. Journal Für Die Reine Und Angewandte Mathematik 645, 155–236 (2010)CrossRefGoogle Scholar
  50. 50.
    Voiculescu, D.: Free analysis questions. I. Duality transform for the coalgebra of X : B. International Mathematics Research Notices 16, 793–822 (2004)Google Scholar
  51. 51.
    Voiculescu, D.: Aspects of free probability. In: XIVth International Congress on Mathematical Physics, pp. 145–157. World Sci. Publ., Hackensack, NJ (2005)Google Scholar

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

Personalised recommendations