On Direct Methods for Lexicographic Min-Max Optimization

  • Włodzimierz Ogryczak
  • Tomasz Śliwiński
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3982)


The approach called the Lexicographic Min-Max (LMM) optimization depends on searching for solutions minimal according to the lex-max order on a multidimensional outcome space. LMM is a refinement of the standard Min-Max optimization, but in the former, in addition to the largest outcome, we minimize also the second largest outcome (provided that the largest one remains as small as possible), minimize the third largest (provided that the two largest remain as small as possible), and so on. The necessity of point-wise ordering of outcomes within the lexicographic optimization scheme causes that the LMM problem is hard to implement. For convex problems it is possible to use iterative algorithms solving a sequence of properly defined Min-Max problems by eliminating some blocked outcomes. In general, it may not exist any blocked outcome thus disabling possibility of iterative Min-Max processing. In this paper we analyze two alternative optimization models allowing to form lexicographic sequential procedures for various nonconvex (possibly discrete) LMM problems. Both the approaches are based on sequential optimization of directly defined artificial criteria. The criteria can be introduced into the original model with some auxiliary variables and linear inequalities thus the methods are easily implementable.


Matrix Game Multiple Objective Linear Programming Multiple Criterion Optimization Individual Objective Function Auxiliary Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Włodzimierz Ogryczak
    • 1
  • Tomasz Śliwiński
    • 1
  1. 1.Institute of Control & Computation EngineeringWarsaw University of TechnologyWarsawPoland

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