Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2305–2323 | Cite as

A modified search direction method for inequality constrained optimization problems using the singular-value decomposition of normalized response gradients

  • Long ChenEmail author
  • Kai-Uwe Bletzinger
  • Armin Geiser
  • Roland Wüchner
Research Paper


Motivated by the applications of the node-based shape optimization problems, where various response evaluations are often considered in constrained optimization, we propose a modified search direction method for optimization design updates. There exist numerous methods for solving constrained optimization problems. The methods of the class active-set strategy try to travel along the active constraints to solve constrained optimization problems. Contrary to the active-set methods, the algorithms of the class interior-point method reach an optimal solution by traversing the interior of the feasible domain. This characteristic is considered to be beneficial for shape optimization because the usual zig-zagging behavior when traveling along the active constraints is avoided. However, the interior-point methods require the solution of a Newton problem in each iteration step, which is generally considered to be difficult for the shape optimization problem. In the present work, we propose a modified search direction method that applies the singular-value decomposition method for the normalized objective and constraint gradients. The modified search direction is a descent direction of the objective function. Using this search direction for the design update in each iteration, we observe that the centrality conditions for the interior-point method are approached iteratively. We are able to achieve a local minimum by traversing the interior of the feasible domain by only using the gradient information of the objective and constraint functions. The results are shown first with analytical 2D problems, and then the results of shape optimization problems with a large number of design variables are discussed. To robustly deal with complex geometries, the Vertex Morphing method is used.


Gradient-based constrained optimization Centrality conditions Singular-value decomposition Shape optimization Vertex Morphing 


Funding information

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the SPP project “Polymorphic uncertainty modelling for the numerical design of structures.”

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Arora JS (2011) Introduction to optimum design, 3rd edn. Academic PressGoogle Scholar
  2. Banerjee S, Roy A (2014) Linear algebra and matrix analysis for statistics. CRC PressGoogle Scholar
  3. Bell R, Koren Y, Volinsky C (2009) Matrix factorization techniques for recommender system. Comput J 8:30–37. Google Scholar
  4. Bertsekas D (1976) On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans Autom Control 21:174–184. MathSciNetCrossRefGoogle Scholar
  5. Bertsekas DP (1999) Nonlinear programming Athena scientific BelmontGoogle Scholar
  6. Bletzinger K-U (2014) A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Struct Multidiscip Optim 49:873–895. MathSciNetCrossRefGoogle Scholar
  7. Bletzinger K-U (2017) Shape optimization, 2nd edn. Stein E, de Borst R, Hughes TJR (eds), WileyGoogle Scholar
  8. Boyd S, Vandenberghe L (2009) Convex optimization. Cambridge University PressGoogle Scholar
  9. Eckart C, Young G (1936) The approximation of the one matrix by another of lower rank. Psychometrika 1:211–218. CrossRefGoogle Scholar
  10. Fletcher R (2013) Practical methods of optimization. WileyGoogle Scholar
  11. Forsgren A, Gill PE, Wright MH (2002) Interior methods for nonlinear optimization. SIAM Rev 44:525–597. MathSciNetCrossRefGoogle Scholar
  12. Frank M, Wolfe P (1956) An algorithm for quadratic programming. Nav Res Logist 3:95–110. MathSciNetCrossRefGoogle Scholar
  13. Gallagher RH, Zienkiewicz OC (1973) Optimum structural design. WileyGoogle Scholar
  14. Golub G, Kahan W (1965) Calculating the singular values and pseudo-inverse of a matrix. SIAM J Numer Anal 2:205–224. MathSciNetzbMATHGoogle Scholar
  15. Golub G, Reinisch C (1970) Singular value decomposition and least squares solutions. Numer Math 14:403–420. MathSciNetCrossRefGoogle Scholar
  16. Golub G, Van Loan CF (2012) Matrix computations, vol 3. JHU PressGoogle Scholar
  17. Hansen PC (1989) Regularization, GSVD and truncated GSVD. BIT 29:491–504. MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hojjat M, Stavropoulou E, Bletzinger K-U (2014) The Vertex Morphing method for node-based shape optimization. Comput Methods Appl Mech Eng 268:494–513. MathSciNetCrossRefGoogle Scholar
  19. Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4:373–395. MathSciNetCrossRefGoogle Scholar
  20. KratosMultiphysics (2019), Accessed on 11 Feb 2019
  21. Mehrotra S (1992) On the implementation of a primal-dual interior point method. SIAM J Optim 2:575–601. MathSciNetCrossRefGoogle Scholar
  22. Militello C, Felippa CA (1991) The first ANDES elements: 9-dof plate bending triangles. Comput Methods Appl Mech Eng 93:217–246. MathSciNetCrossRefGoogle Scholar
  23. Muja M, Lowe D (2014) Scalable nearest neighbor algorithms for high dimensional data. IEEE Trans Pattern Anal Mach Intell 36:2227–2240. CrossRefGoogle Scholar
  24. Najian Asl N, Shayegan S, Geiser A, Hojjat M, Bletzinger K-U (2017) A consistent formulation for imposing packaging constraints in shape optimization using Vertex Morphing parametrization. Struct Multidiscip Optim 56:1507–1519. MathSciNetCrossRefGoogle Scholar
  25. Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. SpringerGoogle Scholar
  26. Pooyang D, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 17:253–297. CrossRefzbMATHGoogle Scholar
  27. Potra FA, Wright SJ (2000) Interior-point methods. J Comput Appl Math 124:281–302. MathSciNetCrossRefGoogle Scholar
  28. Sun W, Yuan Y-X (2006) Optimization theory and methods: nonlinear programming, vol 1. Springer Science & Business MediaGoogle Scholar
  29. Tikhonov AI (1963) Solution of incorrectly formulated problems and the regularization method. Dok Akad Nauk SSSR, 153Google Scholar
  30. Wall ME, Rechtsteiner A, Rocha LM (2003) Singular value decomposition and principal component analysis. In: Berrar DP, Dubitzky W, Granzwo M (eds) A practical approach to microarray data analysis. Springer, Boston, DOI

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technical University of MunichMunichGermany

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