Structural and Multidisciplinary Optimization

, Volume 50, Issue 5, pp 739–753 | Cite as

A globally convergent sequential convex programming using an enhanced two-point diagonal quadratic approximation for structural optimization

  • Seonho Park
  • Seung Hyun Jeong
  • Gil Ho Yoon
  • Albert A. Groenwold
  • Dong-Hoon Choi


In this study, we propose a sequential convex programming (SCP) method that uses an enhanced two-point diagonal quadratic approximation (eTDQA) to generate diagonal Hessian terms of approximate functions. In addition, we use nonlinear programming (NLP) filtering, conservatism, and trust region reduction to enforce global convergence. By using the diagonal Hessian terms of a highly accurate two-point approximation, eTDQA, the efficiency of SCP can be improved. Moreover, by using an appropriate procedure using NLP filtering, conservatism, and trust region reduction, the convergence can be improved without worsening the efficiency. To investigate the performance of the proposed algorithm, several benchmark numerical examples and a structural topology optimization problem are solved. Numerical tests show that the proposed algorithm is generally more efficient than competing algorithms. In particular, in the case of the topology optimization problem of minimizing compliance subject to a volume constraint with a penalization parameter of three, the proposed algorithm is found to converge well to the optimum solution while the other algorithms tested do not converge in the maximum number of iterations specified.


Sequential convex programming (SCP) Diagonal quadratic approximation (DQA) Filter method Conservatism Enhanced two-point diagonal quadratic approximation (eTDQA) 


  1. Alexandrov N M, Dennis J E, Lewis R M, Torczon V (1998) A trust-region framework for managing the use of approximation models in optimization. Struct Optim 15:16–23CrossRefGoogle Scholar
  2. Bendsøe M P, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin; New YorkGoogle Scholar
  3. Bruyneel M, Duysinx P, Fleury C (2002) A family of MMA approximations for structural optimization. Struct Multidiscip O 24:263–276CrossRefGoogle Scholar
  4. Chickermane H, Gea H C (1996) Structural optimization using a new local approximation method. Int J Numer Meth Eng 39:829– 846MathSciNetCrossRefMATHGoogle Scholar
  5. Conn A R, Gould N I M, Toint P L (2000) Trust-region methods. PA, PhiladelphiaCrossRefMATHGoogle Scholar
  6. Duysinx P, Nguyen V H, Bruyneel M, Fleury C (2001) Estimating diagonal second order terms in structural approximations with quasi-Cauchy techniques. In: 4th World congress of structural and multixisciplinary optimization, Dalian ChinaGoogle Scholar
  7. Fadel G M, Riley M F, Barthelemy J M (1990) Two point exponential approximation method for structural optimization. Struct Optim 2:117–124CrossRefGoogle Scholar
  8. Falk J E (1967) Lagrange Multipliers and Nonlinear Programming. J Math Anal Appl 19:141-&Google Scholar
  9. Fletcher R, Gould N I M, Leyffer S, Toint P L, Wachter A (2003) Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming. Siam J Optimiz 13:635–659MathSciNetCrossRefGoogle Scholar
  10. Fletcher R, Leyffer S, Toint P L (2002) On the global convergence of a filter SQP algorithm. Siam J Optimiz 13:44–59MathSciNetCrossRefMATHGoogle Scholar
  11. Fleury C (1979) Structural weight optimization by dual methods of convex programming. Int J Numer Meth Eng 14:1761–1783CrossRefMATHGoogle Scholar
  12. Fleury C (1989) CONLIN: an efficient dual optimizer based on convex approximation concepts. Struct Optim 1:81–89CrossRefGoogle Scholar
  13. Fleury C, Braibant V (1986) Structural Optimization - a New Dual Method Using Mixed Variables. Int J Numer Meth Eng 23:409–428MathSciNetCrossRefMATHGoogle Scholar
  14. Groenwold A A, Etman L F P (2010) A quadratic approximation for structural topology optimization. Int J Numer Meth Eng 82:505–524MathSciNetMATHGoogle Scholar
  15. Groenwold A A, Etman L F P (2010a) On the conditional acceptance of iterates in SAO algorithms based on convex separable approximations. Struct Multidiscip O 42:165–178CrossRefGoogle Scholar
  16. Groenwold A A, Etman L F P, Snyman J A, Rooda J E (2007) Incomplete series expansion for function approximation. Struct Multidiscip O 34:21–40MathSciNetCrossRefMATHGoogle Scholar
  17. Groenwold A A, Etman L F P, Wood D W (2010) Approximated approximations for SAO. Struct Multidiscip O 41:39–56MathSciNetCrossRefMATHGoogle Scholar
  18. Groenwold A A, Wood D W, Etman L F P, Tosserams S (2009) Globally convergent optimization algorithm using conservative convex separable diagonal quadratic approximations. Aiaa J 47:2649–2657CrossRefGoogle Scholar
  19. Kelley C T (1999) Iterative methods for optimization. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  20. Kim J R, Choi D H (2008) Enhanced two-point diagonal quadratic approximation methods for design optimization. Comput Method Appl M 197:846–856CrossRefMATHGoogle Scholar
  21. Park S H, Choi D H (2011) A new convex separable approximation based on two-point diagonal quadratic approximation for large-scale structural design optimization. In: 9th World congress on structural and multidisciplinary optimization. Shizuoka, JapanGoogle Scholar
  22. Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25:493–524CrossRefGoogle Scholar
  23. Svanberg K (1987) The Method of moving asymptotes - a new method for structural optimization. Int J Numer Meth Eng 24:359–373MathSciNetCrossRefMATHGoogle Scholar
  24. Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. Siam J Optimiz 12:555–573MathSciNetCrossRefMATHGoogle Scholar
  25. Vanderplaats G N (1984) Numerical optimization techniques for engineering design: with applications. McGraw-Hill, New YorkMATHGoogle Scholar
  26. Wang L P, Grandhi R V (1994) Efficient safety index calculation for structural reliability-analysis. Comput Struct 52:103–111CrossRefMATHGoogle Scholar
  27. Wang L P, Grandhi R V (1995) Improved two-point function approximation for design optimization. Aiaa J 33:1720– 1727CrossRefMATHGoogle Scholar
  28. Wang L P, Grandhi R V (1996) Multipoint approximations: Comparisons using structural size, configuration and shape design. Struct Optimization 12:177–185CrossRefGoogle Scholar
  29. Wang L P, Grandhi R V, Canfield R A (1996) Multivariate hermite approximation for design optimization. Int J Numer Meth Eng 39:787–803MathSciNetCrossRefMATHGoogle Scholar
  30. Xu G, Yamazaki K, Cheng G D (2000) A new two-point approximation approach for structural optimization. Struct Multidiscip O 20:22–28CrossRefGoogle Scholar
  31. Xu S Q, Grandhi R V (1998) Effective two-point function approximation for design optimization. Aiaa J 36:2269–2275CrossRefGoogle Scholar
  32. Zhang W H, Fleury C (1997) A modification of convex approximation methods for structural optimization. Comput Struct 64:89–95MathSciNetCrossRefMATHGoogle Scholar
  33. Zillober C (1993) A globally convergent version of the method of moving asymptotes. Struct Optimization 6:166–174CrossRefGoogle Scholar
  34. Zillober C (2001) A combined convex approximation-interior point approach for large scale nonlinear programming. Optim Eng 2:51–73MathSciNetCrossRefMATHGoogle Scholar
  35. Zillober C, Schittkowski K, Moritzen K (2004) Very large scale optimization by sequential convex programming. Optim Method Softw 19:103–120MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Seonho Park
    • 1
  • Seung Hyun Jeong
    • 1
  • Gil Ho Yoon
    • 2
  • Albert A. Groenwold
    • 3
  • Dong-Hoon Choi
    • 1
  1. 1.Graduate School of Mechanical EngineeringHanyang UniversitySeoulRepublic of Korea
  2. 2.School of Mechanical EngineeringHanyang UniversitySeoulRepublic of Korea
  3. 3.Department of Mechanical EngineeringUniversity of StellenboschStellenboschSouth Africa

Personalised recommendations