Theory and Algorithms for Finding Knees

Pradyumn Kumar Shukla, Marlon Alexander Braun, Hartmut Schmeck

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)156-170
JournalLecture Notes in Computer Science
Publication statusPublished - 2013


  • knee regions
  • proper Pareto-optimality
  • ordering relations
  • evolutionary algorithms


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