×

zbMATH — the first resource for mathematics

Derivative-free augmented Lagrangian for global optimization: cost minimization in a simplified activated sludge system model. (English) Zbl 1335.90071
Summary: 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 W. Huyer and A. Neumaier [J. Glob. Optim. 14, No. 4, 331–355 (1999; Zbl 0956.90045)]. 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.
MSC:
90C26 Nonconvex programming, global optimization
90C90 Applications of mathematical programming
Software:
OPTIMA; MCS ; MINQ
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramson, MA; Audet, C.; Dennis Jr. J. E., Filter pattern search algorithms for mixed variable constrained optimization problems, SIAM J. Optimiz., 11, 573-594, (2004)
[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)
[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)
[4] Andreani, R.; Birgin, EG; Martinez, JM; Schuverdt, ML, On augmented Lagrangian methods with general lower-level constraints, SIAM J. Optimiz., 18, 1286-1309, (2007) · Zbl 1151.49027
[5] Audet, C.; Dennis, J. E., A pattern search filter method for nonlinear programming without derivatives, SIAM J. Optimiz., 14, 980-1010, (2004) · Zbl 1073.90066
[6] Bartholomew-Biggs, M.: Nonlinear Optimization with Engineering Applications. Springer (2008) · Zbl 1167.90001
[7] Bertsekas, D.: Constrained optimization and Lagrange multiplier methods. Athena Scientific, Belmont (1996)
[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)
[9] Conn, AR; Gould, NIM; Toint, PhL, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM J. Numer. Anal., 28, 545-572, (1991) · Zbl 0724.65067
[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)
[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)
[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)
[13] Espírito Santo, IACP; Fernandes, EMGP; Araújo, MM; Ferreira, EC, On the secondary settler models robustness by simulation, WSEAS Trans. Inf. Sci. Appl., 12, 2323-2330, (2006)
[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)
[15] Güçlü, D.; Dursum, Ş, Artificial neural networks modelling of a large-scale wastewater treatment plant operation, Bioproc. Biosyst. Eng., 33, 1051-1058, (2010)
[16] Hakanen, J.; Miettinen, K.; Sahlstedt, K., Wastewater treatment: New insight provided by interactive multiobjective optimization, Decis. Support Syst., 51, 328-337, (2011)
[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)
[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)
[19] Hooke, R.; Jeeves, TA, Direct search solution of numerical and statistical problems, J. Assoc. Comput., 8, 212-229, (1961) · Zbl 0111.12501
[20] Huyer, W.; Neumaier, A., Global optimization by multilevel coordinate search, J. Glob. Optim., 14, 331-355, (1999) · Zbl 0956.90045
[21] Hydromantis, Inc., Canada, GPS-X V4.1 (2002). http://www.hydromantis.com
[22] Jones, DR; Perttunen, CD; Stuckman, BE, Lipschitzian optimization without the Lipschitz constant, J. Optimiz. Theory App., 79, 157-181, (1993) · Zbl 0796.49032
[23] Kolda, TG; Lewis, RM; Torczon, V., Optimization by direct search: new perspectives on some classical and modern methods, SIAM Rev., 45, 385-482, (2003) · Zbl 1059.90146
[24] Kolda, TG; Lewis, RM; Torczon, V., Stationarity results for generating set search for linearly constrained optimization, SIAM J. Optimiz., 17, 943-968, (2006) · Zbl 1126.90076
[25] Lewis, RM; Torczon, V., Pattern search algorithms for bound constrained minimization, SIAM J. Optimiz., 9, 1082-1099, (1999) · Zbl 1031.90047
[26] Lewis, RM; 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) · Zbl 1011.65030
[27] Luo, H.; Sun, X.; Wu, H., Convergence properties of augmented Lagrangian methods for constrained global optimization, Optim. Method. Softw., 23, 763-778, (2008) · Zbl 1154.90575
[28] Luo, J.: Biegler, L.T.: Dynamic optimization of aeration operations for a benchmark wastewater treatment plant. 18th IFAC Word Congr., 14189-14194 (2011)
[29] Neumaier, A.: MINQ - General Definite and Bound Constrained Indefinite Quadratic Programming. WWW-Document (1998). http://www.mat.univie.ac.at/neum/software/minq/
[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)
[31] Seco, A.; Serralta, J.; Ferrer, J., Biological nutrient removal model no.1 (BNMR1), Water Sci. Technol, 50, 69-78, (2004)
[32] Takȧcs, I.; Patry, GG; Nolasco, D., A dynamic model of the clarification-thickening process, Water Res., 25, 1263-1271, (1991)
[33] Torczon, V., On the convergence of pattern search algorithms, SIAM J. Optimiz., 7, 1-25, (1997) · Zbl 0884.65053
[34] Tyteca, D.; Smeers, Y.; Nyns, EJ, Mathematical modeling and economic optimization of wastewater treatment plants, Crit. Rev. Env. Contr., 8, 1-89, (1977)
[35] Zhang, X.; Zhao, D.; Wang, Z.; Wu, B.; Li, W.; Cheng, SP, Environmental biological model based optimization of activated sludge process, Int. J. Environ. Sci. Te., 6, 69-76, (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.