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Semidefinite optimization (SDO) differs from linear optimization (LO) in that it deals with optimization problems over the cone of symmetric positive semidefinite matrices Sn+ instead of nonnegative vectors. In many cases the objective function is linear and SDO can be interpreted as an extension of LO. It is a branch of convex optimization and contains — besides LO — linearly constrained QP, quadratically constrained QP and — for example — second-order cone programming as special cases.
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References
A. Ben-Tal, A. Nemirovski (2001): Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. SIAM, Philadelphia
M. J. Todd (2001): Semidefinite Optimization. Acta Numerica 10, pp. 515–560
C. Helmberg (2002): Semidefinite Programming. European J. Operational Research 137, pp. 461–482
M. S. Lobo et al. (1998): Applications of Second Order Cone Programming. Linear Algebra and its Applications 284, pp. 193–228
H. Yamashita, H. Yabe, K. Harada (2007): A Primal-Dual Interior Point Method for Nonlinear Semidefinite Programming. Technical Report, Department of Math. Information Science, Tokyo
K.Ċ. Toh (1999): Primal-Dual Path-Following Algorithms for Determinant Maximization Problems with Linear Matrix Inequalities. Computational Optimization and Applications 14, pp. 309–330
L. Vandenberghe, S. Boyd, S. P. Wu (1998): Determinant Maximization with Linear Matrix Inequality Constraints. SIAM J. on Matrix Analysis and Applications 19, pp. 499–533
C. Roos, J. van der Woude (2006): Convex Optimization and System Theory. http://www.isa.ewi.tudelft.nl/~roos/courses/WI4218/
M. Halická, E. de Klerk, C. Roos (2002): On the Convergence of the Central Path in Semidefinite Optimization. SIAM J. Optimization 12, pp. 1090–1099
M. J. Todd (1999): A Study of Search Directions in Primal-Dual Interior-Point Methods for Semidefinite Programming. Optimization Methods and Software 11, pp. 1–46
C. Helmberg et al. (1996): An Interior-Point Method for Semidefinite Programming. SIAM J. Optimization 6, pp. 342–361
M. Kojima, S. Shindoh, S. Hara (1997): A note on the Nesterov-Todd and the Kojima-Shindoh-Hara Search Directions in Semidefinite Programming. SIAM J. Optimization 7, pp. 86–125
R. D. C. Monteiro (1997): Primal-Dual Path-Following Algorithms for Semidefinite Programming. SIAM J. Optimization 7, pp. 663–678
H. Wolkowicz, R. Saigal, L. Vandenberghe (eds.) (2000): Handbook of Semidefinite Programming: Theory, Algorithms and Applications. Kluwer, Boston, Dordrecht, London
E. de Klerk (2002): Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer, Boston, Dordrecht, London
F. Glineur (1998): Pattern Separation via Ellipsoids and Conic Programming. Mémoire de D.E.A. (Master’s Thesis), Faculté Polytechnique de Mons, Belgium
S. Boyd, L. Vandenberghe (2004): Convex Optimization. Cambridge University Press, Cambridge
P. R. Halmos (1993): Finite-Dimensional Vector Spaces. Springer, Berlin, Heidelberg, New York
A. Antoniou, W. Lu (2007): Practical Optimization: Algorithms and Engineering Applications. Springer, Berlin, Heidelberg, New York
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Forst, W., Hoffmann, D. (2010). Semidefinite Optimization. In: Optimization—Theory and Practice. Springer Undergraduate Texts in Mathematics and Technology . Springer, New York, NY. https://doi.org/10.1007/978-0-387-78977-4_7
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