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Algebra of Efficient Sets for Multiobjective Complex Systems

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Abstract

Complex systems are modeled as collections of multiobjective programs representing interacting subsystems of the overall system. Since the calculation of efficient sets of these complex systems is challenging, it is desirable to decompose the overall system into component multiobjective programs, that are more easily solved and then construct the efficient set of the overall system. For some classes of complex systems, algebraic properties of set operations and relations are developed between the efficient set of the overall system and the efficient sets of subproblems. The properties indicate that multiple decomposition and coordination schemes, with varying assumptions regarding the system, may be applied to the same initial system.

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Abbreviations

ℝn :

The set of all n×1 real vectors

X :

The feasible set

\(\mathbb{R}^{p}_{\geqq}\) :

The set of all p×1 real vectors whose components are all nonnegative

\(\mathcal{E}\) :

The efficiency operator, see definition (2)

f○r:

Standard function composition, (f○r)(x)=f(r(x)) for all x∈X

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Box 340975, Clemson, SC, 29634-0975, USA

    Melissa Gardenghi & Margaret M. Wiecek

  2. Department of Applied Economics (Mathematics), University of Málaga, Campus El Ejido s/n, 29071, Málaga, Spain

    Trinidad Gómez & Francisca Miguel

Authors
  1. Melissa Gardenghi
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  2. Trinidad Gómez
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  3. Francisca Miguel
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  4. Margaret M. Wiecek
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Corresponding author

Correspondence to Margaret M. Wiecek.

Additional information

Communicated by H.P. Benson.

M. Gardenghi partially supported by the National Science Foundation Grant CMMI 0621055.

M.M. Wiecek partially supported by the National Science Foundation Grant CMMI 0621055.

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Gardenghi, M., Gómez, T., Miguel, F. et al. Algebra of Efficient Sets for Multiobjective Complex Systems. J Optim Theory Appl 149, 385–410 (2011). https://doi.org/10.1007/s10957-010-9786-y

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