Optimization and Engineering

, Volume 7, Issue 2, pp 201–219 | Cite as

Effective reformulations of the truss topology design problem

  • Michal Kočvara
  • Jiří V. Outrata


We present a new formulation of the truss topology problem that results in unique design and unique displacements of the optimal truss. This is reached by adding an upper level to the original optimization problem and formulating the new problem as an MPCC (Mathematical Program with Complementarity Constraints). We derive optimality conditions for this problem and present several techniques for its numerical solution. Finally, we compare two of these techniques on a series of numerical examples.


Truss topology design Nonlinear programming Mathematical programs with equilibrium constraints 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Achtziger W, Ben-Tal A, Bendsøe M, Zowe J (1992) Equivalent displacement based formulations for maximum strength truss topology design. IMPACT of Computing in Science and Engineering 4:315–345CrossRefMathSciNetGoogle Scholar
  2. Anitescu M, (2000) On solving mathematical programs with complementarity constraints as nonlinear programs. Preprint ANL/MCS-P864-1200, Argonne National Laboratory, Argonne, IL. To appear in SIAM J. OptimizationGoogle Scholar
  3. Ben-Tal A, Bendsøe M (1993) A new iterative method for optimal truss topology design. SIAM J. Optimization 3:322–358CrossRefGoogle Scholar
  4. Ben-Tal A, Nemirovski A (2001) Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM PhiladelphiaGoogle Scholar
  5. Bendsøe M, Sigmund O (2002) Topology optimization. theory, methods and applications. Springer-Verlag, HeidelbergGoogle Scholar
  6. Benson H, Shanno D, Vanderbei R (2002) Interior point methods for nonconvex nonlinear programming: Complementarity comstraints. Technical Report ORFE-02-02, Operations Research and Financial Engineering, Princeton UniversityGoogle Scholar
  7. Byrd R, Hribar M. E, Nocedal, J (1999) An interior point method for large scale nonlinear programming. SIAM J Optimization 9(4):877–900CrossRefMathSciNetGoogle Scholar
  8. Dorn W, Gomory R, Greenberg M (1964) Automatic Design of Optimal Structures. J. de Mechanique 3:25–52Google Scholar
  9. Fletcher R, Leyffer S (2002a) Nonlinear programming without a penalty function. Mathematical Programming 91:239–269CrossRefMathSciNetGoogle Scholar
  10. Fletcher R, Leyffer S (2002b) Solving mathematical program with complementarity constraints as nonlinear programs. Optimization Methods and Software 19(1):15–40CrossRefMathSciNetGoogle Scholar
  11. Fletcher R, Leyffer S, Ralph D, Scholtes S (2002) Local convergence of SQP methods for Mathematical Programs with Equilibrium Constraints. Report NA\209, University of DundeeGoogle Scholar
  12. Gill PE, Murray W, Sounders MA (2002) SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J. Optimization 12:979–1006CrossRefGoogle Scholar
  13. Jarre F, Kočvara M, Zowe J (1998) Interior point methods for mechanical design problems. SIAM J. Optimization 8(4):1084–1107CrossRefGoogle Scholar
  14. Kočvara M, Zibulevsky M, Zowe J (1998) Mechanical design problems with unilateral contact. M2AN Mathematical Modelling and Numerical Analysis 32:255–282Google Scholar
  15. KoČvara M (2000) On the modelling and solving of the truss design problem with global stability constraints. Struct. Multidisc. Optimization 23(3):189–203Google Scholar
  16. KoČvara M, Stingl M (2003) PENNON—A code for convex nonlinear and semidefinite programming. Optimization Methods and Software 18(3):317–333CrossRefMathSciNetGoogle Scholar
  17. Murtagh BA, Saunders MA (1998) MINOS 5.5 User’s Guide. Report SOL 83-20R, Dept of operations research, Stanford UniversityGoogle Scholar
  18. Outrata J (2000) On mathematical programs with complementarity constraints. Optimization Methods and Software 14:117–137MATHMathSciNetGoogle Scholar
  19. Rockafellar RT, Wets RJB (1998) Variational Analysis. Springer, Berlin-HeidelbergGoogle Scholar
  20. Vanderbei RJ, Shanno DF (1999) An interior point algorithm for nonconvex nonlinear programming. Computational Optimization and Applications 13:231–252CrossRefMathSciNetGoogle Scholar
  21. Wächter A, Biegler LT (2004) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Research report, IBM T. J. Watson Research Center, Yorktown, USA. To appear in Mathematical ProgrammingGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic
  2. 2.Faculty of Electrical EngineeringCzech Technical UniversityPraha 6Czech Republic
  3. 3.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicPraha 8Czech Republic

Personalised recommendations