Skip to main content
SpringerLink
Log in

Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns

  • Chapter
  • First Online:
Handbook on Semidefinite, Conic and Polynomial Optimization

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

  • 5669 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 279.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arveson, W.: Subalgebras of C ∗ -algebras. Acta Mathematica 123, 141–224 (1969)

    Article  Google Scholar 

  2. Arveson, W.: Subalgebras of C ∗ -algebras. II. Acta Mathematica 128(3-4), 271–308 (1972)

    Article  Google Scholar 

  3. Arveson, W.: The noncommutative Choquet boundary. Journal of the American Mathematical Society 21(4), 1065–1084 (2008)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  6. Burgdorf, S., Klep, I.: The truncated tracial moment problem. To appear in the Journal of Operator Theory, http://arxiv.org/abs/1001.3679

    Google Scholar 

  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)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  12. Connes, A.: Classification of injective factors. Cases { II}1, { II}∞, { III}λ, λ≠1. Annals of Mathematics 104, 73–115 (1976)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  15. Helton, J.W.: ‘Positive’ noncommutative polynomials are sums of squares. Annals Of Mathematics 156(2), 675–694 (2002)

    Article  Google Scholar 

  16. Helton, J.W., de Oliveira, M.C., Stankus, M., Miller, R.L.: NCAlgebra, 2010 release edition. Available from http://math.ucsd.edu/~ncalg (2010)

  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)

    Article  Google Scholar 

  18. Helton, J.W., Klep, I., McCullough, S.: The matricial relaxation of a linear matrix inequality. Preprint, available from http://arxiv.org/abs/1003.0908

    Google Scholar 

  19. Helton, J.W., Klep, I., McCullough, S.: Proper analytic free maps. Journal of Functional Analysis 260(5), 1476–1490 (2011)

    Article  Google Scholar 

  20. Helton, J.W., Klep, I., McCullough, S., Slinglend, N.: Noncommutative ball maps. Journal of Functional Analysis 257(1), 47–87 (2009)

    Article  Google Scholar 

  21. Helton, J.W., McCullough, S.: Every free basic convex semi-algebraic set has an LMI representation. Preprint, available from http://arxiv.org/abs/0908.4352

    Google Scholar 

  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)

    Article  Google Scholar 

  23. Helton, J.W., McCullough, S.: A positivstellensatz for non-commutative polynomials. Transactions of The American Mathematical Society 356(9), 3721–3737 (2004)

    Article  Google Scholar 

  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)

    Google Scholar 

  25. Helton, J.W., McCullough, S., Putinar, M.: Non-negative hereditary polynomials in a free *-algebra. Mathematische Zeitschrift 250(3), 515–522 (2005)

    Article  Google Scholar 

  26. Helton, J.W., McCullough, S., Putinar, M.: Strong majorization in a free *-algebra. Mathematische Zeitschrift 255(3), 579–596 (2007)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. Helton, J.W., Vinnikov, V.: Linear matrix inequality representation of sets. Communications On Pure And Applied Mathematics 60(5), 654–674 (2007)

    Article  Google Scholar 

  30. Klep, I., Schweighofer, M.: A nichtnegativstellensatz for polynomials in noncommuting variables. Israel Journal of Mathematics 161(1), 17–27 (2007)

    Article  Google Scholar 

  31. Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of Hermitian squares. Advances in Mathematics 217, 1816–1837 (2008)

    Article  Google Scholar 

  32. Klep, I., Schweighofer, M.: Sums of Hermitian squares and the BMV conjecture. Journal of Statistical Physics 133(4), 739–760 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  34. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11(3), 796–817 (2001)

    Article  Google Scholar 

  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. McCullough, S.: Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra and its Applications 326(1-3), 193–203 (2001)

    Article  Google Scholar 

  37. Navascués, M., Pironio, S., Acín, A.: SDP relaxations for non-commutative polynomial optimization. this volume.

    Google Scholar 

  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. Paulsen, V.: Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  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. Pisier, G.: Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  43. Popescu, G.: Noncommutative transforms and free pluriharmonic functions. Advances in Mathematics 220(3), 831–893 (2009)

    Article  Google Scholar 

  44. Popescu, G.: Free holomorphic functions on the unit ball of B(H)n. II. Journal of Functional Analysis 258(5), 1513–1578 (2010)

    Article  Google Scholar 

  45. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42(3), 969–984 (1993)

    Article  Google Scholar 

  46. Schmüdgen, K.: A strict Positivstellensatz for the Weyl algebra. Math. Ann. 331(4), 779–794 (2005)

    Article  Google Scholar 

  47. Schmüdgen, K., Savchuk, Y.: Unbounded induced representations of ∗ -algebras. Preprint, http://arxiv.org/abs/0806.2428

    Google Scholar 

  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. 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)

    Google Scholar 

  50. Voiculescu, D.: Free analysis questions. I. Duality transform for the coalgebra of ∂X : B. International Mathematics Research Notices 16, 793–822 (2004)

    Article  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0112, USA

    J. William Helton

  2. Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Jadranska 21, 1111, Ljubljana, Slovenija, Slovenia

    Igor Klep

  3. Univerza v Mariboru, Fakulteta za naravoslovje in matematiko, Koroška 160, 2000, Maribor, Slovenija, Slovenia

    Igor Klep

  4. Department of Mathematics, University of Florida, 490 Little Hall, Gainesville, FL, 32611-8105, USA

    Scott McCullough

Authors
  1. J. William Helton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Klep
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Scott McCullough
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. William Helton .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Helton, J.W., Klep, I., McCullough, S. (2012). Convexity and Semidefinite Programming in Dimension-Free Matrix Unknowns. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_13

Download citation

Publish with us

Policies and ethics

Search

Navigation