Abstract
Positive semidefinite matrices will be used extensively throughout the book. Therefore we fix notation here and present some basic properties needed later on.
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References
Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002)
Ben-Tal, A., Nemirovski, A.S.: Lectures on Modern Convex Optimization. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)
Borwein, J.M., Wolkowicz, H.: Facial reduction for a cone-convex programming problem. J. Aust. Math. Soc. Ser. A 30 (3), 369–380 (1980/1981)
Brändén, P.: Obstructions to determinantal representability. Adv. Math. 226 (2), 1202–1212 (2011)
Brešar, M., Klep, I.: Noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze. Math. Res. Lett. 16 (4), 605–626 (2009)
Burgdorf, S., Klep, I.: Trace-positive polynomials and the quartic tracial moment problem. C. R. Math. 348 (13–14), 721–726 (2010)
Burgdorf, S., Klep, I.: The truncated tracial moment problem. J. Oper. Theory 68 (1), 141–163 (2012)
Burgdorf, S., Cafuta, K., Klep, I., Povh, J.: The tracial moment problem and trace-optimization of polynomials. Math. Program. 137 (1–2), 557–578 (2013)
Choi, M.-D., Lam, T.Y., Reznick, B.: Sums of squares of real polynomials. In: K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992). Proceedings of Symposia in Pure Mathematics, vol. 58, pp. 103–126. American Mathematical Society, Providence (1995)
Connes, A.: Classification of injective factors. Cases II1, II\(_{\infty },\) III\(_{\lambda },\) \(\lambda \not =1\). Ann. Math. (2) 104 (1), 73–115 (1976)
Curto, R.E., Fialkow, L.A.: Solution of the truncated complex moment problem for flat data. Mem. Am. Math. Soc. 119 (568), x+52 (1996)
Curto, R.E., Fialkow, L.A.: Flat extensions of positive moment matrices: recursively generated relations. Mem. Am. Math. Soc. 136 (648), x+56 (1998)
de Klerk, E.: Aspects of Semidefinite Programming. Applied Optimization, vol. 65. Kluwer Academic, Dordrecht (2002)
Helmberg, C.: Semidefinite Programming for Combinatorial Optimization. Konrad-Zuse-Zentrum für Informationstechnik, Berlin (2000)
Helton, J.W.: “Positive” noncommutative polynomials are sums of squares. Ann. Math. (2) 156 (2), 675–694 (2002)
Helton, J.W., McCullough, S.: A Positivstellensatz for non-commutative polynomials. Trans. Am. Math. Soc. 356 (9), 3721–3737 (2004)
Helton, J.W., McCullough, S.: Every convex free basic semi-algebraic set has an LMI representation. Ann. Math. (2) 176 (2), 979–1013 (2012)
Helton, J.W., Klep, I., McCullough, S.: The convex Positivstellensatz in a free algebra. Adv. Math. 231 (1), 516–534 (2012)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)
Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Noncommutative Function Theory, vol. 199. American Mathematical Society, Providence (2014)
Klep, I., Schweighofer, M.: A nichtnegativstellensatz for polynomials in noncommuting variables. Isr. J. Math. 161, 17–27 (2007)
Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of hermitian squares. Adv. Math. 217 (4), 1816–1837 (2008)
Klep, I., Schweighofer, M.: Sums of hermitian squares and the BMV conjecture. J. Stat. Phys. 133 (4), 739–760 (2008)
Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131, 2nd edn. Springer, New York (2001)
Malick, J., Povh, J, Rendl, F., Wiegele, A.: Regularization methods for semidefinite programming. SIAM J. Optim. 20 (1), 336–356 (2009)
McCullough, S.: Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra Appl. 326 (1–3), 193–203 (2001)
McCullough, S., Putinar, M.: Noncommutative sums of squares. Pac. J. Math. 218 (1), 167–171 (2005)
Mittelman, H.D.: http://plato.asu.edu/sub/pns.html (2015)
Murota, K., Kanno, Y., Kojima, M., Kojima, S.: A numerical algorithm for block-diagonal decomposition of matrix ∗-algebras with application to semidefinite programming. Jpn. J. Ind. Appl. Math. 27 (1), 125–160 (2010)
Netzer, T., Thom, A.: Hyperbolic polynomials and generalized clifford algebras. Discrete Comput. Geom. 51, 802–814 (2014)
Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96 (2, Ser. B), 293–320 (2003)
Peyrl, H., Parrilo, P.A.: Computing sum of squares decompositions with rational coefficients. Theor. Comput. Sci. 409 (2), 269–281 (2008)
Porkolab, L., Khachiyan, L.: On the complexity of semidefinite programs. J. Glob. Optim. 10 (4), 351–365 (1997)
Povh, J., Rendl, F., Wiegele, A.: A boundary point method to solve semidefinite programs. Computing 78, 277–286 (2006)
Powers, V., Scheiderer, C.: The moment problem for non-compact semialgebraic sets. Adv. Geom. 1 (1), 71–88 (2001)
Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. J. Pure Appl. Algebra 127 (1), 99–104 (1998)
Procesi, C.: The invariant theory of n × n matrices. Adv. Math. 19 (3), 306–381 (1976)
Procesi, C., Schacher, M.: A non-commutative real Nullstellensatz and Hilbert’s 17th problem. Ann. Math. 104 (3), 395–406 (1976)
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Ind. Univ. Math. J. 42 (3), 969–984 (1993)
Quarez, R.: Trace-positive non-commutative polynomials. Proc. Am. Math. Soc. 143 (8), 3357–3370 (2015)
Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Program. 77 (2, Ser. B), 129–162 (1997)
Rowen, L.H.: Polynomial Identities in Ring Theory. Pure and Applied Mathematics, vol. 84. Academic, New York (1980)
Scheiderer, C.: Sums of squares of polynomials with rational coefficients. arXiv:1209.2976, to appear in J. Eur. Math. Soc. (2012)
Takesaki, M.: Theory of Operator Algebras. III. Encyclopaedia of Mathematical Sciences, vol. 127. Operator Algebras and Non-commutative Geometry, 8. Springer, Berlin (2003)
Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming. Kluwer, Boston (2000)
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Burgdorf, S., Klep, I., Povh, J. (2016). Selected Results from Algebra and Mathematical Optimization. In: Optimization of Polynomials in Non-Commuting Variables. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33338-0_1
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