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International Conference on Evolutionary Multi-Criterion Optimization

EMO 2013: Evolutionary Multi-Criterion Optimization pp 156–170Cite as

Theory and Algorithms for Finding Knees

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7811))

Abstract

A multi-objective optimization problem involves multiple and conflicting objectives. These conflicting objectives give rise to a set of Pareto-optimal solutions. However, not all the members of the Pareto-optimal set have equally nice properties. The classical concept of proper Pareto-optimality is a way of characterizing good Pareto-optimal solutions. In this paper, we metrize this concept to induce an ordering on the Pareto-optimal set. The use of this metric allows us to define a proper knee region, which contains solutions below a user-specified threshold metric. We theoretically analyze past definitions of knee points, and in particular, reformulate a commonly used nonlinear program, to achieve convergence results. Additionally, mathematical properties of the proper knee region are investigated. We also develop two multi-objective evolutionary algorithms towards finding proper knees and present simulation results on a number of test problems.

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Authors and Affiliations

  1. Institute AIFB, Karlsruhe Institute of Technology, Karlsruhe, D-76128, Germany

    Pradyumn Kumar Shukla, Marlon Alexander Braun & Hartmut Schmeck

Authors
  1. Pradyumn Kumar Shukla
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  2. Marlon Alexander Braun
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  3. Hartmut Schmeck
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Editor information

Editors and Affiliations

  1. Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, S1 3JD, Sheffield, UK

    Robin C. Purshouse

  2. Department of Automatic Control Systems and Engineering, University of Sheffield, Mappin Street, S1 3JD, Sheffield, UK

    Peter J. Fleming

  3. Department of Informatics Engineering, University of Coimbra, Pólo II, Pinhal de Marrocos, 3030-290, Coimbra, Portugal

    Carlos M. Fonseca

  4. Department of Economics and Business, University of Catania, Corso Italia 55, 95129, Catania, Italy

    Salvatore Greco

  5. Unilever Research & Development Port Sunlight, Quarry Road East, CH63 3JW, Merseyside, UK

    Jane Shaw

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© 2013 Springer-Verlag Berlin Heidelberg

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Shukla, P.K., Braun, M.A., Schmeck, H. (2013). Theory and Algorithms for Finding Knees. In: Purshouse, R.C., Fleming, P.J., Fonseca, C.M., Greco, S., Shaw, J. (eds) Evolutionary Multi-Criterion Optimization. EMO 2013. Lecture Notes in Computer Science, vol 7811. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37140-0_15

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  • DOI: https://doi.org/10.1007/978-3-642-37140-0_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-37139-4

  • Online ISBN: 978-3-642-37140-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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