Abstract
In this first chapter we give an introduction and outline of the topics from the book. We also introduce basic notions and results from linear algebra, convexity theory and real algebraic geometry.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, 2002)
A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization. Analysis, Algorithms, and Engineering Applications. MPS/SIAM Series on Optimization (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001)
G. Blekherman, P.A. Parrilo, R.R. Thomas (eds.), Semidefinite Optimization and Convex Algebraic Geometry. MOS-SIAM Series on Optimization, vol. 13 (Society for Industrial and Applied Mathematics (SIAM)/Mathematical Optimization Society, Philadelphia, 2013)
J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36 (Springer-Verlag, Berlin, 1998)
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, vol. 15 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994)
S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)
G.B. Dantzig, M.N. Thapa, Linear Programming. Springer Series in Operations Research, vol. 1 (Springer-Verlag, New York, 1997)
G.B. Dantzig, M.N. Thapa, Linear Programming. Springer Series in Operations Research, vol. 2 (Springer-Verlag, New York, 2003). Theory and extensions.
E. de Klerk, M. Laurent, On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems. SIAM J. Optim. 21(3), 824–832 (2011)
M. Grötschel, L. Lovász, A. Schrijver, Polynomial algorithms for perfect graphs, in Topics on Perfect Graphs. North-Holland Math. Stud., vol. 88 (North-Holland, Amsterdam, 1984), pp. 325–356
J.W. Helton, J. Nie, Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim. 20(2), 759–791 (2009)
J.W. Helton, J. Nie, Semidefinite representation of convex sets. Math. Program. A 122(1), 21–62 (2010)
J.W. Helton, V.Vinnikov, Linear matrix inequality representation of sets. Commun. Pure Appl. Math. 60(5), 654–674 (2007)
J.B. Lasserre, Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (electronic) (2000/2001)
J.B. Lasserre, Convex sets with semidefinite representation. Math. Program. 120(2, Ser. A), 457–477 (2009)
M. Laurent, S. Poljak, On a positive semidefinite relaxation of the cut polytope. Linear Algebra Appl. 223/224, 439–461 (1995). Special issue honoring Miroslav Fiedler and Vlastimil Pták
P.D. Lax, Differential equations, difference equations and matrix theory. Commun. Pure Appl. Math. 11, 175–194 (1958)
A.S. Lewis, P.A. Parrilo, M.V. Ramana, The Lax conjecture is true. Proc. Am. Math. Soc. 133(9), 2495–2499 (electronic) (2005)
M. Marshall, Positive Polynomials and Sums of Squares. Mathematical Surveys and Monographs, vol. 146 (American Mathematical Society, Providence, 2008)
Y. Nesterov, A. Nemirovski, Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, vol. 13 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994)
P.A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D Thesis, 2000
G. Pataki, On the facial structure of cone-lp’s and semidefinite programs. Management Science Research Report, MSRR-595, 1994
A. Prestel, C.N. Delzell, Positive Polynomials. Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2001)
M. Ramana, A.J. Goldman, Some geometric results in semidefinite programming. J. Global Optim. 7(1), 33–50 (1995)
R.T. Rockafellar, Convex Analysis. Princeton Mathematical Series, vol. 28 (Princeton University Press, Princeton, 1970)
C. Scheiderer, Semidefinite representation for convex hulls of real algebraic curves. SIAM J. Appl. Algebra Geom. 2(1), 1–25 (2018)
C. Scheiderer, Spectrahedral shadows. SIAM J. Appl. Algebra Geom. 2(1), 26–44 (2018)
M. Todd, Semidefinite optimization. Acta Numer. 10, 515–560 (2001)
L. Vandenberghe, S. Boyd, Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)
R. Webster, Convexity. Oxford Science Publications (The Clarendon Press/Oxford University Press, New York, 1994)
Wolfram Research, Inc., Mathematica, Version 13.2 (Wolfram Research, Inc., Champaign, 2022)
H. Wolkowicz, R. Saigal, L. Vandenberghe (eds.), Handbook of Semidefinite Programming. Theory, Algorithms, and Applications. International Series in Operations Research & Management Science, vol. 27 (Kluwer Academic Publishers, Boston, 2000)
G.M. Ziegler, Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152 (Springer-Verlag, New York, 1995)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Netzer, T., Plaumann, D. (2023). Introduction and Preliminaries. In: Geometry of Linear Matrix Inequalities. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-26455-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-031-26455-9_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-26454-2
Online ISBN: 978-3-031-26455-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)