Skip to main content
SpringerLink
Log in

Constructing a Pareto front approximation for decision making

  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

An approach to constructing a Pareto front approximation to computationally expensive multiobjective optimization problems is developed. The approximation is constructed as a sub-complex of a Delaunay triangulation of a finite set of Pareto optimal outcomes to the problem. The approach is based on the concept of inherent nondominance. Rules for checking the inherent nondominance of complexes are developed and applying the rules is demonstrated with examples. The quality of the approximation is quantified with error estimates. Due to its properties, the Pareto front approximation works as a surrogate to the original problem for decision making with interactive methods.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bezerkin VE, Kamenev GK, Lotov AV (2006) Hybrid adaptive methods for approximating a nonconvex multidimensional Pareto frontier. Comput Math Math Phys 46: 1918–1931

    Article  MathSciNet  Google Scholar 

  • Boissonnant J-D (1984) Geometric structures for three-dimensional shape representation. ACM Trans Graph 3: 266–286

    Article  Google Scholar 

  • Caratheodory C (1913) Bedingt Konvergente Reihen und Konvexe Systeme. J für die reine und angewandte Mathematik 143: 128–175

    Google Scholar 

  • Coello Coello CA, Van Veldhuizen DA, Lamont GB (2007) Evolutionary algorithms for solving multi-objective problems. Springer, New York

    MATH  Google Scholar 

  • Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, Chichester

    MATH  Google Scholar 

  • Deb K, Thiele L, Laumanns M, Zitzler E (2002) Scalable multi-objective optimization test problems. E-Commer Technol IEEE Int Conf 1: 825–830

    Google Scholar 

  • Edelsbrunner H (1987) Algorithms in combinatorial geometry. Springer, New York

    MATH  Google Scholar 

  • Edelsbrunner H (1998) Shape reconstruction with Delaunay complex. In: Lucchesi CL, Moura AV (eds) LATIN’98: theoretical informatics. Springer, Berlin, pp 119–132

    Chapter  Google Scholar 

  • Edelsbrunner H, Shah NR (1994) Triangulating topological spaces. In: SCG ’94: proceedings of the tenth annual symposium on computational geometry. ACM, New York, pp. 285–292

  • Efremov RV, Kamenev GK (2009) Properties of a method for polyhedral approximation of the feasible criterion set in convex multiobjective problems. Ann Oper Res 166: 271–279

    Article  MathSciNet  MATH  Google Scholar 

  • Eskelinen P, Miettinen K, Klamroth K, Hakanen J (2010) Pareto navigator for interactive nonlinear multiobjective optimization. OR Spectr 23: 211–227

    Article  MathSciNet  Google Scholar 

  • Fortune S (1997) Voronoi diagrams and Delaunay triangulations. In: Goodman JE, O’Rourke J Handbook of discrete and computational geometry. CRC Press, Boca Raton

  • Fukuda K (2004) Polyhedral computation FAQ. Swiss Federal Institute of Technology, http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html

  • George P-L, Borouchaki H (1998) Delaunay triangulation and meshing: application to finite elements. Hermes, Paris

    MATH  Google Scholar 

  • Goel T, Vaidyanathan R, Haftka RT, Shyy W, Queipo NV, Tucker K (2007) Response surface approximation of Pareto optimal front in multi-objective optimization. Comput Methods Appl Mech Eng 196: 879–893

    Article  MATH  Google Scholar 

  • Goodman, JE, O’Rourke, J (eds) (1997) Discrete and computational geometry. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Grünbaum B (1967) Convex polytopes. Interscience Publishers, London

    MATH  Google Scholar 

  • Hartikainen M, Miettinen K, Wiecek MM (2011) Pareto front approximations for decision making with inherent non-dominance. In: Yong S, Shouyang W, Gang K, Wallenius J (eds) The proceedings of the MCDM2009 conference, lecture notes in economics and mathematical systems. Springer, Berlin. To Appear

  • Hasenjäger M, Sendhoff B (2005) Crawling along the Pareto front: tales from the practice. In: The 2005 IEEE congress on evolutionary computation (IEEE CEC 2005). IEEE Press, Piscataway, NJ, pp 174–181

  • Keeney RL, Raiffa H (1993) Decisions with multiple objectives. Cambridge University Press, Cambridge

    Google Scholar 

  • Laukkanen T, Tveit T-M, Ojalehto V, Miettinen K, Fogelholm C-J (2010) An interactive multi-objective approach to heat exchanger network synthesis. Comput Chem Eng 34: 943–952

    Article  Google Scholar 

  • Lotov AV, Bushenkov VA, Kamenev GA (2004) Interactive decision maps. Kluwer, Boston

    MATH  Google Scholar 

  • Luque M, Ruiz F, Miettinen K (2011) Global formulation for interactive multiobjective optimization. OR Spectr. 33: 27–48

    Article  MATH  Google Scholar 

  • Martin J, Bielza C, Insua DR (2005) Approximating nondominated sets in continuous multiobjective optimization problems. Nav Res Logist 52: 469–480

    Article  MATH  Google Scholar 

  • McMullen P (1970) The maximum number of faces of a convex polytope. Mathematika 17: 179–184

    Article  MathSciNet  MATH  Google Scholar 

  • Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer, Boston

    MATH  Google Scholar 

  • Miettinen K, Mäkelä M (1995) Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization 34: 231–246

    Article  MathSciNet  MATH  Google Scholar 

  • Miettinen K, Mäkelä M (2000) Interactive multiobjective optimization system WWW-NIMBUS on the internet. Comput Oper Res 27: 709–723

    Article  MATH  Google Scholar 

  • Miettinen K, Mäkelä MM (2006) Synchronous approach in interactive multiobjective optimization. Eur J Oper Res 170: 909–922

    Article  MATH  Google Scholar 

  • Miettinen K, Ruiz F, Wierzbicki A (2008) Introduction to multiobjective optimization: interactive approaches. In: Branke J, Deb K, Miettinen K, Slowinski R (eds) Multiobjective optimization: interactive and evolutionary approaches. Springer, Berlin, pp 27–57

    Google Scholar 

  • Monz M (2006) Pareto navigation—interactive multiobjective optimization and its applications in radiotherapy planning. Ph.D. thesis, University of Kaiserslautern

  • Rajan VT (1994) Optimality of the Delaunay triangulation in \({\mathbb R^d}\). Discret Comput Geom 12: 189–202

    Article  MathSciNet  MATH  Google Scholar 

  • Ruzika S, Wiecek MM (2005) Approximation methods in multiobjective programming. J Optim Theory Appl 126: 473–501

    Article  MathSciNet  MATH  Google Scholar 

  • Sawaragi Y, Nakayama H, Tanino T (1985) Theory of multiobjective optimization. Academic Press, Orlando

    MATH  Google Scholar 

  • Schandl B, Klamroth K, Wiecek MM (2001) Norm-based approximation in bicriteria programming. Comput Optim Appl 20: 23–42

    Article  MathSciNet  MATH  Google Scholar 

  • Schandl B, Klamroth K, Wiecek MM (2002) Norm-based approximation in multicriteria programming. Comput Math Appl 44: 925–942

    Article  MathSciNet  MATH  Google Scholar 

  • Shenton D, Cendes Z (1985) Three-dimensional finite element mesh generation using Delaunay tesselation. IEEE Trans Magn 21: 2535–2538

    Article  Google Scholar 

  • Steuer RE (1986) Multiple criteria optimization: theory, computation and application. Wiley, New York

    MATH  Google Scholar 

  • Steuer RE (1989) The Tchebycheff procedure of interactive multiple objective programming. In: Karpak B, Zionts S (eds) Multiple criteria decision making and risk analysis using micro computers. Springer, Berlin

    Google Scholar 

  • Steuer RE, Choo E-U (1983) An interactive weighted Tchebycheff procedure for multiple objective programming. Math Program 26: 326–344

    Article  MathSciNet  MATH  Google Scholar 

  • Vegter G (1997) Computational topology. In: Goodman JE, O’Rourke J (eds) Handbook of discrete and computational geometry. CRC Press, Boca Raton, pp 719–742

    Google Scholar 

  • Wierzbicki A (1986) On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spectr 8: 73–87

    MathSciNet  MATH  Google Scholar 

  • Yu PL, Zeleny M (1975) The set of all nondominated solutions in linear cases and a multicriteria simplex method. J Math Anal Appl 49: 430–468

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Markus Hartikainen, Kaisa Miettinen & Margaret M. Wiecek

Authors
  1. Markus Hartikainen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kaisa Miettinen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Margaret M. Wiecek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Hartikainen.

Additional information

On sabbatical leave from Department of Mathematical Sciences, Clemson University, Clemson, SC, USA. This research was partly supported by the Academy of Finland grant number 128495.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hartikainen, M., Miettinen, K. & Wiecek, M.M. Constructing a Pareto front approximation for decision making. Math Meth Oper Res 73, 209–234 (2011). https://doi.org/10.1007/s00186-010-0343-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-010-0343-0

Keywords

Search

Navigation