Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty

  • Y. Kanno
  • I. Takewaki


A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of the external forces based on the info-gap model, the maximization of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has a global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without difficulty.


Robust optimization info-gap model semidefinite programs structural optimization successive linearization methods 


  1. 1.
    TSOMPANAKIS, Y., and PAPADRAKAKIS, M., Large-Scale Reliability-Based Structural Optimization, Structural and Multidisciplinary Optimization, Vol. 26, pp. 429–440, 2004.CrossRefGoogle Scholar
  2. 2.
    KHARMANDA, G., OLHOFF, N., MOHAMED, A., and LEMAIRE, M., Reliability-Based Topology Optimization, Structural and Multidisciplinary Optimization, Vol. 26, pp. 295–307, 2004.CrossRefGoogle Scholar
  3. 3.
    CHOI, K. K., TU, J., and PARK, Y. H., Extensions of Design Potential Concept for Reliability-Based Design Optimization to Nonsmooth and Extreme Cases, Structural and Multidisciplinary Optimization, Vol. 22, pp. 335–350, 2001.CrossRefGoogle Scholar
  4. 4.
    JUNG, D. H., and LEE, B. C., Development of a Simple and Efficient Method for Robust Optimization, International Journal for Numerical Methods in Engineering, Vol. 53, pp. 2201–2215, 2002.CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    DOLTSINIS, I., and KANG, Z., Robust Design of Structures Using Optimization Methods, Computer Methods in Applied Mechanics and Engineering, Vol. 193, pp. 2221–2237, 2004.CrossRefMATHGoogle Scholar
  6. 6.
    BEN-HAIM, Y., and ELISHAKOFF, I., Convex Models of Uncertainty in Applied Mechanics, Elsevier, New York, NY, 1990.MATHGoogle Scholar
  7. 7.
    PANTELIDES, C. P., and GANZERLI, S., Design of Trusses under Uncertain Loads using Convex Models, ASCE Journal of Structural Engineering, Vol. 124, pp. 318–329, 1998.CrossRefGoogle Scholar
  8. 8.
    BEN-TAL, A., and NEMIROVSKI, A., Robust Optimization: Methodology and Applications, Mathematical Programming, Vol. 92B, pp. 453–480, 2002.CrossRefMathSciNetGoogle Scholar
  9. 9.
    BEN-TAL, A., and NEMIROVSKI, A., Robust Truss Topology Optimization via Semidefinite Programming, SIAM Journal on Optimization, Vol. 7, pp. 991–1016, 1997.CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    KOČVARA, M., ZOWE, J., and NEMIROVSKI, A., Cascading: An Approach to Robust Material Optimization, Computers and Structures, Vol. 76, pp. 431–442, 2000.CrossRefGoogle Scholar
  11. 11.
    HAN, J. S., and KWAK, B. M., Robust Optimization using a Gradient Index: MEMS Applications, Structural and Multidisciplinary Optimization, Vol. 27, pp. 469–478, 2004.CrossRefGoogle Scholar
  12. 12.
    CALAFIORE, G., and EL GHAOUI, L., Ellipsoidal Bounds for Uncertain Linear Equations and Dynamical Systems, Automatica, Vol. 40, pp. 773–787, 2004.CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    BEN-HAIM, Y., Information-Gap Decision Theory, Academic Press, London, UK, 2001.MATHGoogle Scholar
  14. 14.
    KANNO, Y., and TAKEWAKI, I., Robustness Analysis of Trusses with Separable Load and Structural Uncertainties, International Journal of Solids and Structures, Vol. 43, pp. 2646–2669, 2006.Google Scholar
  15. 15.
    WOLKOWICZ, H., SAIGAL, R., and VANDENBERGHE, L., Editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Dordrecht, Netherlands, 2000.Google Scholar
  16. 16.
    JARRE, F., Some Aspects of Nonlinear Semidefinite Programming, System Modeling and Optimization: 20th IFIP Conference on System Modeling and Optimization, Edited by E. W. Sachs and R. Tichatschke, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 55–69, 2003.Google Scholar
  17. 17.
    FUKUSHIMA, M., TAKAZAWA, K., OHSAKI, S., and IBARAKI, T., Successive Linearization Methods for Large-Scale Nonlinear Programming Problems, Japan Journal of Industrial and Applied Mathematics, Vol. 9, pp. 117–132, 1992.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    KANZOW, C., NAGEL, C., KATO, H., and FUKUSHIMA, M., Successive Linearization Methods for Nonlinear Semidefinite Programs, Computational Optimization and Applications, Vol. 31, pp. 251–273, 2005.CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    KOJIMA, M., and TUNÇEL, L., Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets, SIAM Journal on Optimization, Vol. 10, pp. 750–778, 2000.CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    SIMO, J. C., and HUGHES, T. J. R., Computational Inelasticity, Springer-Verlag, New York, NY, 1998.MATHGoogle Scholar
  21. 21.
    BEN-TAL, A., and NEMIROVSKI, A., Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, SIAM, Philadelphia, Pennysylvania, 2001.Google Scholar
  22. 22.
    OHSAKI, M., FUJISAWA, K., KATOH, N., and KANNO, Y., Semidefinite Programming for Topology Optimization of Truss under Multiple Eigenvalue Constraints, Computer Methods in Applied Mechanics and Engineering, Vol. 180, pp. 203–217, 1999.CrossRefMATHGoogle Scholar
  23. 23.
    KANNO, Y., OHSAKI, M., and KATOH, N., Sequential Semidefinite Programming for Optimization of Framed Structures under Multimodal Buckling Constraints, International Journal of Structural Stability and Dynamics, Vol. 1, pp. 585–602, 2001.CrossRefGoogle Scholar
  24. 24.
    STURM, J. F., Using SeDuMi 1.02, a MATLAB Toolbox for Optimization over Symmetric Cones, Optimization Methods and Software, Vol. 11/12, pp. 625–653, 1999.Google Scholar
  25. 25.
    THE MATH WORKS, Using MATLAB, The Math Works, Natick, Massachusetts, 2002.Google Scholar
  26. 26.
    KANNO, Y., and TAKEWAKI, I., Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainties, Kyoto University, BGE Research Report 04–05, 2004 (revised 2005); available at∼bge/RR/.
  27. 27.
    FUJISAWA, K., KOJIMA, M., and NAKATA, K., Exploiting Sparsity in Primal-Dual Interior-Point Methods for Semidefinite Programming, Mathematical Programming, Vol. 79, pp. 235–253, 1997.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Y. Kanno
    • 1
  • I. Takewaki
    • 2
  1. 1.Department of Mathematical InformaticsThe University of TokyoTokyoJapan
  2. 2.Department of Urban and Environmental EngineeringKyoto UniversityKyotoJapan

Personalised recommendations