Abstract
10 years ago or so Bill Helton introduced me to some mathematical problems arising from semidefinite programming. This paper is a partial account of what was and what is happening with one of these problems, including many open questions and some new results.
Mathematics Subject Classification. Primary: 14M12, 90C22; secondary: 14P10, 52A20.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris. Geometry of Algebraic Curves: Volume I. Springer, New York, 1985.
J. Backelin, J. Herzog, and H. Sanders. Matrix factorizations of homogeneous polynomials. Algebra some current trends (Varna, 1986), pp. 1–33. Lecture Notes in Math., 1352, Springer, Berlin, 1988.
J.A. Ball and V. Vinnikov. Zero-pole interpolation for matrix meromorphic functions on an algebraic curve and transfer functions of 2D systems. Acta Appl. Math. 45:239–316, 1996.
J.A. Ball and V. Vinnikov. Zero-pole interpolation for meromorphic matrix functions on a compact Riemann surface and a matrix Fay trisecant identity. Amer. J. Math. 121:841–888, 1999.
A. Beauville. Determinantal hypersurfaces. Mich. Math. J. 48:39–64, 2000.
P. Brändén. Obstructions to determinantal representability. Advances in Math., to appear (arXiv:1004.1382).
H.H. Bauschke, O. Güler, A.S. Lewis, and H.S. Sendov. Hyperbolic polynomials and convex analysis. Canad. J. Math. 53:470–488, 2001.
A. Buckley and T. Košir. Determinantal representations of smooth cubic surfaces. Geom. Dedicata 125:115–140, 2007.
R.J. Cook and A.D. Thomas. Line bundles and homogeneous matrices. Quart. J. Math. Oxford Ser. (2) 30:423–429, 1979.
A. Degtyarev and I. Itenberg. On real determinantal quartics. Preprint (arXiv:1007.3028).
L.E. Dickson. Determination of all general homogeneous polynomials expressible as determinants with linear elements. Trans. Amer. Math. Soc. 22:167–179, 1921.
A. Dixon. Note on the reduction of a ternary quartic to a symmetrical determinant. Proc. Cambridge Phil. Soc. 11:350-351, 1900–1902 (available at http://www.math.bgu.ac.il/kernerdm ).
I. Dolgachev. Classical Algebraic Geometry: A Modern View. Cambridge Univ. Press, to appear (available at http://www.math.lsa.umich.edu).
B.A. Dubrovin. Matrix finite zone operators. Contemporary Problems of Mathematics (Itogi Nauki i Techniki) 23, pp. 33–78 (1983) (Russian).
J.D. Fay. Theta Functions On Riemann Surfaces. Lecture Notes in Math. 352, Springer, Berlin, 1973.
D. Eisenbud. Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260:35-64, 1980.
L. Gårding. Linear hyperbolic partial differential equations with constant coefficients. Acta Math. 85:2–62, 1951.
L. Gårding. An inequality for hyperbolic polynomials. J. Math. Mech. 8:957–965, 1959.
B. Grenet, E. Kaltofen, P. Koiran, and N. Portier. Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits. Randomization, Relaxation, and Complexity in Polynomial Equation Solving (edited by L. Gurvits, P. Pebay, and J.M. Rojas), pp. 61–96. Contemporary Mathematics 556, Amer. Math. Soc., 2011 (arXiv:1007.3804).
P. Griffiths and J. Harris. On the Noether–Lefschetz theorem and some remarks on codimension two cycles. Math. Ann. 271:31–51, 1985.
O. Güler. Hyperbolic polynomials and interior point methods for convex programming. Mathematics of Operations Research 22:350–377, 1997.
R. Hartshorne. Ample subvarieties of algebraic varieties. Lecture Notes in Math., 156, Springer-Verlag, Heidelberg, 1970.
J.W. Helton and S.A. McCullough. Every convex free basic semi-algebraic set has an LMI representation. Preprint [http://arxiv.org/abs/0908.4352]
J.W. Helton, S. McCullough, M. Putinar, and V. Vinnikov. Convex matrix inequalities versus linear matrix inequalities. IEEE Trans. Automat. Control 54:952–964, 2009.
J.W. Helton and J. Nie. Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim. 20:759–791, 2009.
J.W. Helton and J. Nie. Semidefinite representation of convex sets. Math. Program. 122:21–64, 2010.
J.W. Helton, S.A. McCullough, and V. Vinnikov. Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240:105–191, 2006.
J.W. Helton and V. Vinnikov. Linear Matrix Inequality Representation of Sets. Comm. Pure Appl. Math. 60:654–674, 2007.
D. Henrion. Detecting rigid convexity of bivariate polynomials. Linear Algebra Appl. 432:1218–1233, 2010.
J. Herzog, B. Ulrich, and J. Backelin. Linear Cohen-Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71:187–202, 1991.
S.-G. Hwang. Cauchy’s interlace theorem for eigenvalues of Hermitian matrices. Amer. Math. Monthly 111:157–159, 2004.
D. Kerner and V. Vinnikov. On the determinantal representations of singular hypersurfaces in Pn. Advances in Math., to appear (arXiv:0906.3012).
D. Kerner and V. Vinnikov. On decomposability of local determinantal representations of hypersurfaces. Preprint (arXiv:1009.2517).
M.G. Krein and M.A. Naimark. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Kharkov, 1936. English translation (by O. Boshko and J.L. Howland): Lin. Mult. Alg. 10:265–308, 1981.
J.B. Lasserre. Convex sets with semidefinite representation. Math. Program. 120:457477, 2009.
P. Lax. Differential equations, difference equations and matrix theory. Comm. Pure Appl. Math. 11:175–194, 1958.
S. Lefschetz. L’analysis situs et la géometrie algébrique. Gauthier-Villars, Paris, 1924.
A.S. Lewis, P.A. Parrilo, and M.V. Ramana. The Lax conjecture is true. Proc. Amer. Math. Soc. 133:2495–2499, 2005.
A. Nemirovskii. Advances in convex optimization: conic programming. Proceedings of the International Congress of Mathematicians (ICM) (Madrid, 2006), Vol. I (Plenary Lectures), pp. 413–444. Eur. Math. Soc., Zurich, 2007 (available at http://www.icm2006.org/proceedings).
Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics, 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994.
T. Netzer, D. Plaumann, and M. Schweighofer. Exposed faces of semidefinitely representable sets. SIAM J. Optim. 20:1944–1955, 2010.
T. Netzer, D. Plaumann, and A. Thom. Determinantal representations and the Hermite matrix. Preprint (arXiv:1108.4380).
T. Netzer, A. Thom: Polynomials with and without determinantal representations. Preprint (arXiv:1008.1931).
W. Nuij. A note on hyperbolic polynomials. Math. Scand. 23:69–72, 1968.
P. Parrilo and B. Sturmfels. Minimizing polynomial functions. Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001), pp. 83–99. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, Amer. Math. Soc., Providence, RI, 2003.
D. Plaumann, B. Sturmfels, and C. Vinzant. Quartic Curves and Their Bitangents. Preprint (arXiv:1008.4104).
D. Plaumann, B. Sturmfels, and C. Vinzant. Computing Linear Matrix Representations of Helton-Vinnikov Curves. This volume (arXiv:1011.6057).
R. Quarez. Symmetric determinantal representations of polynomials. Preprint (http://hal.archives-ouvertes.fr/hal-00275615_vl).
M. Ramana and A.J. Goldman. Some geometric results in semidefinite programming. J. Global Optim. 7:33–50, 1995.
J. Renegar. Hyperbolic programs, and their derivative relaxations. Found. Comput. Math. 6:59–79, 2006.
R.E. Skelton, T. Iwasaki, and K.M. Grigoriadis. A Unified Algebraic Approach to Linear Control Design. Taylor & Francis, 1997.
L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Rev. 38:49–95, 1996.
V. Vinnikov. Complete description of determinantal representations of smooth irreducible curves. Lin. Alg. Appl. 125:103–140, 1989.
V. Vinnikov. Self-adjoint determinantal representions of real plane curves. Math. Ann. 296:453–479, 1993.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
To Bill Helton, on the occasion of his 65th birthday
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Vinnikov, V. (2012). LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_23
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0411-0_23
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0410-3
Online ISBN: 978-3-0348-0411-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)