Derivative-Free Augmented Lagrangian for Global Optimization: Cost Minimization in a Simplified Activated Sludge System Model

  • Isabel Espirito Santo
  • Roman Denysiuk
  • Edite M.G.P. Fernandes


A methodology for finding the optimal values of the decision variables from an efficient simplified mathematical model of an activated sludge system is addressed in this paper. The work herein presented arises in a wastewater treatment plant design context, where investment and operational costs are to be minimized and computational effort is to be reduced. To achieve the best design, a non-linear optimization solution method based on an augmented Lagrangian approach is proposed. At each iteration, a subproblem is globally solved by a derivative-free recursive branching technique, known as the multilevel coordinate search algorithm of Huyer and Neumaier [20]. The presented technique has been shown to work quite well when solving the herein proposed non-convex and non-smooth constrained optimization model. The numerical results show the reliability of the obtained solutions at a reduced computational cost.


WWTP design Activated sludge system Cost minimization Augmented Lagrangian Multilevel coordinate search 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramson, M.A., Audet, C., Dennis Jr. J.E.: Filter pattern search algorithms for mixed variable constrained optimization problems. SIAM J. Optimiz. 11, 573–594 (2004)Google Scholar
  2. 2.
    Afonso, P.N.C.M.: Modelação matemática de reactores biológicos no tratamento terciário de efluentes. Ph.D. Thesis (in portuguese), Universidade do Porto (2001)Google Scholar
  3. 3.
    Alex, J., Benedetti, L., Copp, J., Gernaey, K., Jeppsson, U., Nopens, I., Pons, M., Rosen, C., Steyer, J., Vanrolleghem, P.: Benchmark simulation model no. 1 (BSM1). Technical Report, IWA Taskgroup pn Benchmarking of Control Strategies for WWTPs (2008)Google Scholar
  4. 4.
    Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optimiz. 18, 1286–1309 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Audet, C., Dennis Jr. J.E.: A pattern search filter method for nonlinear programming without derivatives. SIAM J. Optimiz. 14, 980–1010 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bartholomew-Biggs, M.: Nonlinear Optimization with Engineering Applications. Springer (2008)Google Scholar
  7. 7.
    Bertsekas, D.: Constrained optimization and Lagrange multiplier methods. Athena Scientific, Belmont (1996)Google Scholar
  8. 8.
    Clara, N.: Neural networks complemented with genetic algorithms and fuzzy systems for predicting nitrogenous effluent variables in wastewater treatment plants. WSEAS Trans. Syst. 7, 695–705 (2008)Google Scholar
  9. 9.
    Conn, A.R., Gould, N.I.M., Toint, Ph.L.: A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545–572 (1991)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Elias, A., Ibarra, G., Ormazabal, J., Murgia, I., Zugazti, P.: ADM: A model for water treatment in an anaerobic biological reactor. In: Brebbia, C.A. (ed.) Development and Application of Computer Techniques to Environmental Studies VI, Wessex Institute of Technology, United Kingdom & P. Zannetti, Failure Analysis Associates Inc, California (1996)Google Scholar
  11. 11.
    Ekama, G.A., Barnard, J.L., Günthert, F.W., Krebs, P., McCrquodale, J.A., Parker, D.S., Wahlberg, E.J.: Secondary settling tanks: Theory, modelling, design and operation, Technical Report No. 6, IAWQ - International Association on Water Quality (1997)Google Scholar
  12. 12.
    Espírito Santo, I.A.C.P., Fernandes, E.M.G.P.: Simplified model for the activated sludge system: WWTP cost minimization via an augmented Lagrangian pattern search method. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Numerical Analysis and Applied Mathematics: ICNAAM 2010, AIP Conference Proceedings ISBN: 978-0-7354-0834-0, Vol. 1281 pp. 971-974 (2010)Google Scholar
  13. 13.
    Espírito Santo, I.A.C.P., Fernandes, E.M.G.P., Araújo, M.M., Ferreira, E.C.: On the secondary settler models robustness by simulation. WSEAS Trans. Inf. Sci. Appl. 12, 2323–2330 (2006)Google Scholar
  14. 14.
    Espírito Santo, I.A.C.P., Fernandes, E.M.G.P., Araújo, M.M., Ferreira, E.C.: An augmented Lagrangian pattern search method for optimal WWTP designs. In: Proceedings of the ICOSSSE ’07, pp. 313–318 (2007)Google Scholar
  15. 15.
    Güçlü, D., Dursum, Ş.: Artificial neural networks modelling of a large-scale wastewater treatment plant operation. Bioproc. Biosyst. Eng. 33, 1051–1058 (2010)CrossRefGoogle Scholar
  16. 16.
    Hakanen, J., Miettinen, K., Sahlstedt, K.: Wastewater treatment: New insight provided by interactive multiobjective optimization. Decis. Support Syst. 51, 328–337 (2011)CrossRefGoogle Scholar
  17. 17.
    Henze, M., Grady, C.P.L., Gujer, W., Marais, G.V.R., Matsuo, T.: Activated Sludge Model No. 1. Scientific and Technical Report, Vol. 1. IWA Publishing, London (1987)Google Scholar
  18. 18.
    Henze, M., Gujer, W., Mino, T., Van Loosdrecht, M.C.M.: Activated Sludge Models: ASM1, ASM2, ASM2d and ASM3. Scientific and Technical Report, Vol. 9. IWA Publishing, London (2000)Google Scholar
  19. 19.
    Hooke, R., Jeeves, T.A.: Direct search solution of numerical and statistical problems. J. Assoc. Comput. 8, 212–229 (1961)CrossRefMATHGoogle Scholar
  20. 20.
    Huyer, W., Neumaier, A.: Global optimization by multilevel coordinate search. J. Glob. Optim. 14, 331–355 (1999)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Hydromantis, Inc., Canada, GPS-X V4.1 (2002).
  22. 22.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optimiz. Theory App. 79, 157–181 (1993)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45, 385–482 (2003)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Kolda, T.G., Lewis, R.M., Torczon, V.: Stationarity results for generating set search for linearly constrained optimization. SIAM J. Optimiz. 17, 943–968 (2006)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lewis, R.M., Torczon, V.: Pattern search algorithms for bound constrained minimization. SIAM J. Optimiz. 9, 1082–1099 (1999)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Lewis, R.M., Torczon, V.: A globally convergent augmented Lagrangian pattern search algorithm for optimization with general constraints and simple bounds. SIAM J. Optimiz. 12, 1075–1089 (2002)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Luo, H., Sun, X., Wu, H.: Convergence properties of augmented Lagrangian methods for constrained global optimization. Optim. Method. Softw. 23, 763–778 (2008)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Luo, J.: Biegler, L.T.: Dynamic optimization of aeration operations for a benchmark wastewater treatment plant. 18th IFAC Word Congr., 14189–14194 (2011)Google Scholar
  29. 29.
    Neumaier, A.: MINQ - General Definite and Bound Constrained Indefinite Quadratic Programming. WWW-Document (1998).
  30. 30.
    Otterpohl, R., Rolfs, T., Londong, J.: Optimizing operation of wastewater treatment plants by offline and online computer simulation. Water Sci. Technol 30, 165–174 (1994)Google Scholar
  31. 31.
    Seco, A., Serralta, J., Ferrer, J.: Biological nutrient removal model no.1 (BNMR1). Water Sci. Technol 50, 69–78 (2004)Google Scholar
  32. 32.
    Takȧcs, I., Patry, G.G., Nolasco, D.: A dynamic model of the clarification-thickening process. Water Res. 25, 1263–1271 (1991)CrossRefGoogle Scholar
  33. 33.
    Torczon, V.: On the convergence of pattern search algorithms. SIAM J. Optimiz. 7, 1–25 (1997)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Tyteca, D., Smeers, Y., Nyns, E.J.: Mathematical modeling and economic optimization of wastewater treatment plants. Crit. Rev. Env. Contr. 8, 1–89 (1977)CrossRefGoogle Scholar
  35. 35.
    Zhang, X., Zhao, D., Wang, Z., Wu, B., Li, W., Cheng, S.P.: Environmental biological model based optimization of activated sludge process. Int. J. Environ. Sci. Te. 6, 69–76 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Isabel Espirito Santo
    • 1
  • Roman Denysiuk
    • 1
  • Edite M.G.P. Fernandes
    • 1
  1. 1.Algoritmi Research CentreUniversity of Minho, Campus de GualtarBragaPortugal

Personalised recommendations