Abstract
This paper is an extended abstract of a survey talk given at the IFIP TC7 Conference in Trier, July 2001. We consider linear and nonlinear semidefinite programming problems and concentrate on selected aspects that are relevant for understanding dual barrier methods. The paper is aimed at graduate students to highlight some issues regarding smoothness, regularity, and computational complexity without going into details.
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35699-0_19
Chapter PDF
Similar content being viewed by others
References
F. Alizadeh, “Combinatorial Optimization with Interior Point Methods and Semidefinite Matrices” PhD Thesis, University of Minnesota (1991).Optimierung, Operations Research, Spieltheorie Birkhäuser Verlag (2001).
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, Volume 15 of Studies in Applied Mathematics SIAM Philadelphia, PA, (1994).
A.R. Conn, N.I.M. Gould and Ph.L. Toint, Trust-Region Methods MPS/SIAM Series on Optimization, SIAM, Philadelphia, (2000).
A. Forsgren, “Optimality conditions for nonconvex semidefinite programming”, Mathematical Programming 88 (2000), 105–128.
R.W. Freund and F. Jarre, “An Extension of the Positive Real Lemma to Descriptor Systems” Report 00/3–09, Scientific Computing Interest Group, Bell Labs, Lucent Technologies, (2000)
M. Fukuda and M. Kojima, “Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem”, Computational Optimization and Applications Vol. 19, No. 1, pp. 79–105, (2001).
M.X. Goemans and D.P. Williamson, “.878-Approximation Algorithm for MAX CUT and MAX 2SAT”, in ACM Symposium on Theory of Computing (STOC), (1994).
C. Helmberg, F. Rendi, R.J. Vanderbei, “An Interior-Point Method for Semidefinite Programming” SIAM J. Optim. 6(2):342–361 (1996).
M.W. Hirsch and S. Smale Differential equations, dynamical systems, and linear algebra Acad. Press, New York, (1974).
B.W. Hirschfeld and F. Jarre, “Complexity Issues of Smooth Local Minimization”, Technical Report, Universität Düsseldorf, in preparation (2001). “Interior-Point Algorithms for Classes of Convex Programs” in T. Terlaky ed. Interior Methods of Mathematical Programming Kluwer (1996).
F. Jarre, “A QQP-Minimization Method for Semidefinite and Smooth Nonconvex Programs”, Technical Report, University of Düsseldorf, Germany, to appear in revised form in Optimization and Engineering (2001).
F. Jarre and M.A. Saunders, “A Practical Interior-Point Method for Convex Programming”, SIAM J. Optim. 5 (1) pp. 149–171 (1995).
H.T. Jongen and A. Ruiz Jhones, “Nonlinear Optimization: On the Min-Max Digraph and Global Smoothing”, in A. Ioffe, S. Reich, I. Shafrir eds Calculus of Variations and Differential Equations Chapman Hall/CRC Research Notes in Mathematics Series, Vol 410, CRC Press, ( UK) LLC, pp. 119–135, (1999).
M. Kocvara and M. Stingl, “Augmented Lagrangian Method for Semidefinite Programming” forthcoming report, Institute of Applied Mathematics, University of Erlangen-Nuremberg (2001).
F. Leibfritz, “A LMI-based algorithm for designing suboptimal static/output feedback controllers”, SIAM Journal on Control and Optimization, Vol. 39, No. 6, pp. 1711–1735, (2001).
Y.E. Nesterov, Talk given at the Conference on Semidefinite Optimization, ZIB Berlin, (1997).
Y.E. Nesterov, “Semidefinite Relaxation and Nonconvex Quadratic Optimization”, Optimization Methods and Software 9, pp. 141–160, (1998).
Y.E. Nesterov and A.S. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming, SIAM Publications, SIAM Philadelphia, PA, (1994).
G. Pataki, “The Geometry of Semidefinite Programming”, in H. Wolkowicz, R. Saigal, L. Vandenberghe eds Handbook of Semidefinite Programming: Theory, Algorithms and Applications Kluwer’s International Series (2000).
C. Scherer, “Lower bounds in multi-objective H2/Hœ problems”, Proc. 38th IEEE Conf. Decision and Control, Arizona, Phoenix (1999).
A. Shapiro and K. Scheinberg, “Duality and Optimality Conditions”, in H. Wolkowicz, R. Saigal, L. Vandenberghe eds Handbook of Semidefinite Programming: Theory, Algorithms and Applications Kluwer’s International Series (2000).
Ph.L. Toint, “Some Numerical Result Using a Sparse Matrix Updating Formula in Unconstrained Optimization”, Mathematics of Computation,vol. 32(143), pp. 839851, (1978).
R.J. Vanderbei, “LOQO User’s Manual — Version 3.10”. Report SOR 97–08, Princeton University, Princeton, NJ 08544, (1997, revised 10/06/98).
R.J. Vanderbei, H. Benson and D. Shanno, “Interior-Point Methods for Nonconvex Nonlinear Programming. Filter Methods and Merit Functions”, Report ORFE 0006, Princeton University, Princeton, NJ 08544, (2000).
S.A. Vavasis, “Black-Box Complexity of Local Minimization”, SIAM J. Optim. 3 (1) pp. 60–80, (1993).
S.J. Wright and J. Nocedal, Numerical Optimization, Springer Verlag, (1999).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 IFIP International Federation for Information Processing
About this paper
Cite this paper
Jarre, F. (2003). Some Aspects of Nonlinear Semidefinite Programming. In: Sachs, E.W., Tichatschke, R. (eds) System Modeling and Optimization XX. CSMO 2001. IFIP — The International Federation for Information Processing, vol 130. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35699-0_3
Download citation
DOI: https://doi.org/10.1007/978-0-387-35699-0_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-6669-1
Online ISBN: 978-0-387-35699-0
eBook Packages: Springer Book Archive