Augmented Lagrangian and Nonlinear Semidefinite Programs

  • X. X. Huang
  • X. Q. Yang
  • K. L. Teo
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)


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.

Key words

Semidefinite programming augmented Lagrangian duality exact penalization convergence stationary point 


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  1. [1]
    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.CrossRefMathSciNetGoogle Scholar
  2. [2]
    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.Google Scholar
  3. [3]
    Bertsekas, D. P., “Constrained Optimization and Lagrangian Multiplier Methods”. Academic Press, New York, 1982.Google Scholar
  4. [4]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    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.Google Scholar
  8. [8]
    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.CrossRefGoogle Scholar
  9. [9]
    Forsgren, A., “Optimality conditions for nonconvex semidefinite programming”. Mathematical Programming, Ser. A., Vol. 88, 2000, pp. 105–128.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Fares, B., Noll, D. and Apkarian, P., “Robust control via sequential semidefinite programming”. SIAM J. Control and Optimi., Vol. 40, 2002, pp. 1791–1820.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Ghaoui, L. E. and Niculescu, S. I., “Advances in Linear Matrix Inequality Methods in Control”. Advances in Design Control, SIAM, Philadelphia, 2000.Google Scholar
  12. [12]
    Horn, R. A. and Johnson, C. R., “Topics in Matrix Analysis”. Cambridge University Press, Cambridge, 1991.zbMATHGoogle Scholar
  13. [13]
    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.Google Scholar
  14. [14]
    Jarre, F., “An interior point method for semidefinite programs”. Optimization and Engineering, Vol. 1, 2000, pp. 347–372.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Mosheyev, L. and Zibulevsky, M., “Penalty/barrier multiplier algorithm for semidefinite programming”. Optimization Methods and Software, Vol. 13, 2000, pp. 235–261.zbMATHMathSciNetGoogle Scholar
  17. [17]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    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.Google Scholar
  19. [19]
    Rockafellar, R. T., “Augmented Lagrange multiplier functions and duality in nonconvex programming”, SIAM Jou. on Control and Optimization, Vol. 12, 1974, pp. 268–285.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Rockafellar, R. T., “Lagrange multipliers and optimality”. SIAM Review, Vol. 35, 1993, pp. 183–238.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    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.Google Scholar
  22. [22]
    Todd, M., “Semidefinite Optimization”. Acta Numerica, Vol. 10, 2001, pp. 515–560.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    Vandenberghe, L. and Boyd, S., “Semidefinite programming”. SIAM Review, Vol. 38, 1996, pp. 49–95.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    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.Google Scholar
  25. [25]
    Ye, Y., “Interior Point Algorithms: Theory and Analysis”. John Wiley & Son, New York, 1997.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • X. X. Huang
    • 1
  • X. Q. Yang
    • 1
  • K. L. Teo
    • 1
  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloon, Hong KongP.R. of China

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