Abstract
In this paper, we introduce an augmented Lagrangian for nonlinear semidefinite programs. Some basic properties of the augmented Lagrangian such as differentiabilty, monotonicity and convexity, are discussed. Necessary and sufficient conditions for a strong duality property and an exact penalty representation in the framework of augmented Lagrangian are derived. Under certain conditions, it is shown that any limit point of a sequence of stationary points of augmented Lagrangian problems is a Karuh, Kuhn-Tucker (for short, KKT) point of the original semidefinite program.
This work is supported by a Postdoctoral Fellowship of The Hong Kong Polytechnic University.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bel-Tal, A., Jarre, F., Kocvara, M., Nemirovski and Zowe, J., “Optimal design of trusses under a nonconvex global buckling constraints”. Optimization and Engineering, Vol. 1, 2000, pp. 189–213.
Benson, H. Y. and Vanderbei, R. J., “Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming”. Mathematical Programming, Ser. B, Vol. 93, 2002.
Bertsekas, D. P., “Constrained Optimization and Lagrangian Multiplier Methods”. Academic Press, New York, 1982.
Bonnans, J. F., Cominetti, R. and Shapiro, A., “Second order optimality conditions based on second order tangent sets”. SIAM Jou. Optimization, Vol. 9, 1999, pp. 466–492.
Burer, S., Monteiro, R. D. C. and Zhang, Y., “Solving a class of semidefinite programs via nonlinear programming”. Mathematical Programming, Ser. A., Vol. 93, 2002, pp. 97–122.
Burer, S., Monteiro, R. D. C. and Zhang, Y., “Interior-point algorithms for semidefinite programming based on a nonlinear formulation”. Computational Optimization and Applications, Vol. 22, 2002, pp. 49–79.
Chen, X., Qi, H. D. and Tseng, P., “Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complementarity constraints”. SIAM J. Optimization. To appear.
Fan, K., “On a theorem of Wely concerning eigenvalues of linear transformations”. I., Proc. Nat. Acad. Sci. U. S. A., Vol. 35, 1949, pp. 652–655.
Forsgren, A., “Optimality conditions for nonconvex semidefinite programming”. Mathematical Programming, Ser. A., Vol. 88, 2000, pp. 105–128.
Fares, B., Noll, D. and Apkarian, P., “Robust control via sequential semidefinite programming”. SIAM J. Control and Optimi., Vol. 40, 2002, pp. 1791–1820.
Ghaoui, L. E. and Niculescu, S. I., “Advances in Linear Matrix Inequality Methods in Control”. Advances in Design Control, SIAM, Philadelphia, 2000.
Horn, R. A. and Johnson, C. R., “Topics in Matrix Analysis”. Cambridge University Press, Cambridge, 1991.
Jarre, F., “Convex analysis on symmetric matrices”. In “Handbook of Semidefinite Programming, Theory, Algorithms and Applications”, H. Wolkowicz, R. Saigal and Vandenberghe (eds), Kluwer Academic Publishers, 2000.
Jarre, F., “An interior point method for semidefinite programs”. Optimization and Engineering, Vol. 1, 2000, pp. 347–372.
Kanzow, C. and Nagel, C, “Semidefinite programs: new search directions, smoothing-type methods, and numerical results”. SIAM Jou. Optimization, Vol. 13, 2002, pp. 1–23.
Mosheyev, L. and Zibulevsky, M., “Penalty/barrier multiplier algorithm for semidefinite programming”. Optimization Methods and Software, Vol. 13, 2000, pp. 235–261.
Overton, M. L. and Womersley, R. S., “Second derivatives for optimizing eigenvalues of symmetric matrices”. SIAM J. Matrix Analysis and Applications, Vol. 16, 1995, pp. 697–718.
Ringertz, U. T., “Eigenvalues in optimal structural design”. In: Biegler, L. T., Coleman, T. F. Conn, A. R. and Santosa, F. N. (eds), “Large Scale Optimization and Applications, Part I: Optimization in Inverse Problems and Design”, Vol. 92 of the IMA Volumes in Mathematics and its Applications, pp. 135–149, Springer, New York, 1997. of the IMA Volumes in Mathematics and its Applications, Springer, New York, 1997, pp. 135–149.
Rockafellar, R. T., “Augmented Lagrange multiplier functions and duality in nonconvex programming”, SIAM Jou. on Control and Optimization, Vol. 12, 1974, pp. 268–285.
Rockafellar, R. T., “Lagrange multipliers and optimality”. SIAM Review, Vol. 35, 1993, pp. 183–238.
Shapiro, A., “First and second order analysis of nonlinear semidefinite programs”. Mathematical Programming, Ser. B., Vol. 77, 1997, pp. 301–320. National University of Singapore, Singapore, 2002.
Todd, M., “Semidefinite Optimization”. Acta Numerica, Vol. 10, 2001, pp. 515–560.
Vandenberghe, L. and Boyd, S., “Semidefinite programming”. SIAM Review, Vol. 38, 1996, pp. 49–95.
Wolkowicz, H., Saigal, R. and Vandenberghe, L. (eds), “Handbook of semidefinite programming, theory, algorithms and applications”. International Series in Operations Research and Management Science, Vol. 27, Kluwer Academic Publishers, Boston, MA, 2000.
Ye, Y., “Interior Point Algorithms: Theory and Analysis”. John Wiley & Son, New York, 1997.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Huang, X.X., Yang, X.Q., Teo, K.L. (2005). Augmented Lagrangian and Nonlinear Semidefinite Programs. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol 79. Springer, Boston, MA. https://doi.org/10.1007/0-387-24276-7_32
Download citation
DOI: https://doi.org/10.1007/0-387-24276-7_32
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-24209-5
Online ISBN: 978-0-387-24276-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)