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Extracting from the relaxed for large-scale semi-continuous variable nondominated frontiers

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Abstract

Because of size and covariance matrix problems, computing much of anything along the nondominated frontier of a large-scale (1000–3000 securities) portfolio selection problem with semi-continuous variables is a task that has not previously been achieved. But given (a) the speed at which the nondominated frontier of a classical portfolio problem can now be computed and (b) the possibility that there might be overlaps between the nondominated frontier of the classical problem and that of the same problem but with semi-continuous variables, the paper shows how considerable amounts of the nondominated frontier of a large-scale mean-variance portfolio selection problem with semi-continuous variables can be computed in very little time.

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Notes

  1. Also known as the “efficient frontier”.

  2. For more about the \(\varepsilon \)-constraint method than utilized here, see Mavrotas [19] and Miettinen [20].

  3. All times in this paper are from an i7-2720 2.20 GHz computer. Sample sizes are 10 throughout.

  4. In the pension fund arena alone, the 300th largest pension fund has assets in excess of $11 billion (Towers Watson, The World’s 300 Largest Pension Funds—Year End 2012, www.towerswatson.com).

  5. A word about the bottommost hyperbolic “segment” of the nondominated frontier: In portfolio selection there is the minimum standard deviation boundary as shown in Sharpe [22]. It is entirely constructed out of hyperbolic segments, and the upper portion of this boundary is the nondominated frontier, that is, from the global minimum standard deviation point upward. With the global minimum standard deviation point likely falling within the relative interior of one of the hyperbolic segments of the minimum standard deviation boundary, the bottommost hyperbolic segment of the nondominated frontier will normally be observed to be a subset of this generally larger hyperbolic segment.

  6. Or stop in the case of the lower endpoint of the bottommost hyperbolic segment, see footnote 5.

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Acknowledgments

The authors would like to acknowledge helpful comments from the referees.

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Authors and Affiliations

  1. Department of Finance, University of Georgia, Athens, GA, 30602-6253, USA

    Ralph E. Steuer

  2. Munich Re, 80802, Munich, Germany

    Markus Hirschberger

  3. College of Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Kalyanmoy Deb

Authors
  1. Ralph E. Steuer
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  2. Markus Hirschberger
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  3. Kalyanmoy Deb
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Correspondence to Ralph E. Steuer.

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Steuer, R.E., Hirschberger, M. & Deb, K. Extracting from the relaxed for large-scale semi-continuous variable nondominated frontiers. J Glob Optim 64, 33–48 (2016). https://doi.org/10.1007/s10898-015-0305-4

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