Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Incomplete series expansion for function approximation

Abstract

We present an incomplete series expansion (ISE) as a basis for function approximation. The ISE is expressed in terms of an approximate Hessian matrix, which may contain second, third, and even higher order “main” or diagonal terms, but which excludes “interaction” or off-diagonal terms. From the ISE, a family of approximation functions may be derived. The approximation functions may be based on an arbitrary number of previously sampled points, and any of the function and gradient values at suitable previously sampled points may be enforced when deriving the approximation functions. When function values only are enforced, the storage requirements are minimal. However, irrespective of the conditions enforced, the approximate Hessian matrix is a sparse diagonal matrix. In addition, the resultant approximations are separable. Hence, the proposed approximation functions are very well-suited for use in gradient-based sequential approximate optimization requiring computationally expensive simulations; a typical example is structural design problems with many design variables and constraints. We derived a wide selection of approximations from the family of ISE approximating functions; these include approximations based on the substitution of reciprocal and exponential intervening variables. A comparison with popular approximating functions previously proposed illustrates the accuracy and flexibility of the new family of approximation functions. In fact, a number of popular approximating functions previously proposed for structural optimization applications derive from our ISE.

This is a preview of subscription content, log in to check access.

References

  1. Alexandrov NM, Dennis JE, Lewis RM, Torczon V (1998) A trust region framework for managing the use of approximation models in optimization. Struct Optim 15:16–23

  2. Barthelemy JFM, Haftka RT (1993) Approximation concepts for optimum structural design—a review. Struct Optim 5:129–144

  3. Barthold F-J, Stander N, Stein E (1996) Performance comparison of SAM and SQP methods for structural shape optimization. Struct Optim 11:102–112

  4. Bruyneel M, Duysinx P, Fleury C (2002) A family of MMA approximations for structural optimization. Struct Multidisc Optim 24:263–276

  5. Canfield RA (1990) High-quality approximation of eigenvalues in structural optimization. AIAA J 28(6):1116–1122

  6. Canfield RA (2004) Multipoint cubic surrogate function for sequential approximate optimization. Struct Multidisc Optim 27:326–336

  7. Chickermane H, Gea HC (1996) Structural optimization using a new local approximation method. Int J Numer Methods Eng 39:829–846

  8. Conn AR, Gould NIM, Toint PL (2000) Trust-region methods MPS/SIAM series on optimization. SIAM, Philadelphia, PA

  9. English TM (1996) Evaluation of evolutionary and genetic operators: no free lunch. In: Fogel LJ, Angeline PJ, Baeck T (eds) Evolutionary programming V. Proceedings of the fifth annual conference on evolutionary programming, San Diego, CA, USA, 29 February–2 March 1996 , pp 163–169

  10. Etman LFP, Groenwold AA, Rooda JE (2006) Sequential approximate optimization in an NLP filter framework. In: Proceedings of the 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, Virginia, USA, September 2006. Paper no. AIAA-2006-7078

  11. Fadel GM, Riley MF, Barthelemy JM (1990) Two point exponential approximation method for structural optimization. Struct Optim 2:117–124

  12. Fleury C (1989) Efficient approximation concepts using second order information. Int J Numer Methods Eng 28:2041–2058

  13. Fleury C, Braibant V (1986) Structural optimization: a new dual method using mixed variables. Int J Numer Methods Eng 23:409–428

  14. Groenwold AA, Etman LFP (2006a) Duality in convex non-linear multipoint approximations with diagonal approximate Hessian matrices deriving from incomplete series expansions. In: Proceedings of the 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, Virginia, USA, September 2006. Paper no. AIAA-2006-7090

  15. Groenwold AA, Etman LFP (2006b) Optimality criterion methods and sequential approximate optimization in the classical topology layout problem. In: Proceedings of the eighth international conference on computational structures technology, Las Palmas de Gran Canaria, Spain, September 2006. Paper no. CST2006/2005/000216

  16. Groenwold AA, Etman LFP, Snyman JA, Rooda JE (2005) Incomplete series expansion for function approximation. In: Proceedings of the sixth world congress on structural and multidisciplinary optimization, Rio de Janeiro, Brazil, May 2005.

  17. Haftka RT, Gürdal Z (1992) Elements of structural optimization. Kluwer, Dordrecht

  18. Haftka RT, Shore CP (1979) Approximation method for combined thermal/structural design. NASA TP-1428

  19. Haftka RT, Nachlas JA, Watson LA, Rizzo T, Desai R (1987) Two point constraint approximation in structural optimization. Comput Methods Appl Mech Eng 60:289–301

  20. Prasad B (1983) Explicit constraint approximation forms in structural optimization. part 1: analyses and projections. Comput Methods Appl Mech Eng 40:1–26

  21. Ringertz UT (1988) On methods for discrete structural optimization. Eng Optim 13:47–64

  22. Salajegheh E (1997) Optimum design of plate structures using three-point approximation. Struct Optim 13:142–147

  23. Snyman JA, Hay AM (2001) The spherical quadratic steepest descent (SQSD) method for unconstrained minimization with no explicit line searches. Comput Math Appl 42:169–178

  24. Snyman JA, Hay AM (2002) The Dynamic-Q optimization method: an alternative to SQP? Comput Math Appl 44:1589–1598

  25. Snyman JA, Stander N (1994) New successive approximation method for optimum structural design. AIAA J 32:1310–1315

  26. Starnes Jr. JH, Haftka RT (1979) Preliminary design of composite wings for buckling, stress and displacement constraints. J Aircr 16:564–570

  27. Sunar M, Belegundu AD (1991) Trust region methods for structural optimization using exact second order sensitivity. Int J Numer Methods Eng 32:275–293

  28. Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

  29. Svanberg K (1995) A globally convergent version of MMA without linesearch. In: Rozvany GIN, Olhoff N (eds) Proceedings of the first world congress on structural and multidisciplinary optimization, Goslar, Germany, pp 9–16

  30. Wang L, Grandhi RV (1994) Efficient safety index calculations for structural reliability analysis. Comput Struct 52:103–111

  31. Wang L, Grandhi RV (1995) Improved two-point function approximation for design optimization. AIAA J 33:1720–1727

  32. Wolpert D, Macready W (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82

  33. Xu S, Grandhi RV (1998) Effective two-point function approximation for design optimization. AIAA J 36:2269–2275

  34. Xu G, Yamazaki K, Cheng GD (2000) A new two-point approximation approach for structural optimization. Struct Multidisc Optim 20:22–28

Download references

Author information

Correspondence to Albert A. Groenwold.

Additional information

Based on the similarly named paper presented at the Sixth World Congress on Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, May 2005

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Groenwold, A.A., Etman, L.F.P., Snyman, J.A. et al. Incomplete series expansion for function approximation. Struct Multidisc Optim 34, 21–40 (2007). https://doi.org/10.1007/s00158-006-0070-6

Download citation

Keywords

  • Nonlinear function approximation
  • Sequential approximate optimization (SAO)
  • Incomplete series expansion (ISE)