Matrix cubes parameterized by eigenvalues

SIAM Journal on Matrix Analysis and Applications

Volumn 31, Issue 2, 2009, Pages 755-766

  Nie, Jiawang ^a   Sturmfels, Bernd ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Algebraic degree; K ellipse; Linear matrix inequality (lmi); Matrix cube; Semidefinite programming (sdp); Tensor product; Tensor sum

Indexed keywords

ALGEBRAIC DEGREES; MATRIX; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE PROGRAMMING (SDP); TENSOR PRODUCTS;

ALGEBRA; EIGENVALUES AND EIGENFUNCTIONS; GEOMETRY; LINEAR MATRIX INEQUALITIES; MATHEMATICAL PROGRAMMING; SET THEORY; TOPOLOGY;

TENSORS;

EID: 72449210145     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/080722606     Document Type: Article

References (16)

1. S. BASU, R. POLLACK, AND M.-F. ROY, Algorithms in Real Algebraic Geometry, Springer-Verlag, Berlin, 2006.
2. S. BOYD, L. EL GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, SIAM Stud. Appl. Math. 15, SIAM, Philadelphia, 1994.
3. A. BEN-TAL AND A. NEMIROVSKI, Robust convex optimization, Math. Oper. Res., 23 (1998), pp. 769-805.
4. A. BEN-TAL, A. NEMIROVSKI, AND C. ROOS, Extended matrix cube theorems with applications to μ-theory in control, Math. Oper. Res., 28 (2003), pp. 497-523.
5. B.-D. CHEN AND S. LALL, Degree bounds for polynomial verification of the matrix cube problem, in Proceedings of the 45th IEEE Conference on Decision and Control, SanDiego, CA, 2006, IEEE Press, Piscataway NJ, 2006, pp. 4405-4410.
6. J. W. HELTON AND J. NIE, Semidefinite representation of convex sets, Math. Program., to appear.
7. J. W. HELTON AND J. NIE, Sufficient and necessary conditions for semidefinite representability of convex hulls and sets, SIAM J. Optim., 20 (2009), pp. 759-791.
8. J. W. HELTON AND V. VlNNIKOV, Linear matrix inequality representation of sets, Comm. Pure Appl. Math., 60 (2007), pp. 654-674.
9. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1994.
10. S. LANG, Algebra, 3rd ed., Springer-Verlag, New York, 2000.
11. A. NEMIROVSKI, The Matrix Cube Problem: Approximations and Applications, lecture notes, 2001, available online at http://www2.isye.gatech.edu/-nemirovs/.
12. J. NIE, P. PARRILO, AND B. STURMFELS, Semidefinite representation of the k-ellipse, in Algorithms in Algebraic Geometry, IMA Vol. Math. Appl. 146, A. Dickenstein, F.-O. Schreyer, and A. Sommese, eds., Springer, New York, 2008, pp. 117-132.
13. J. NIE, K. RANESTAD, AND B. STURMFELS, The algebraic degree of semidefinite programming, Math. Program., to appear.
14. M. RAMANA AND A. J. GOLDMAN, Some geometric results in semidefinite programming,J. Global Optim., 7 (1995), pp. 33-50.
15. J. F. STURM, Sedumi 1.02, A MATLAB toolbox for optimization over symmetric cones,Optim. Methods Softw., 11/12 (1999), pp. 625-653.
16. H. WOLKOWICZ, R. SAIGAL, AND L. VANDENBERGHE, EDS., Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Dordrecht, he Netherlands, 2000.