The impact of the existence of multiple adjustable robust solutions

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Abstract

In this note we show that multiple solutions exist for the production-inventory example in the seminal paper on adjustable robust optimization in Ben-Tal et al. (Math Program 99(2):351–376, 2004). All these optimal robust solutions have the same worst-case objective value, but the mean objective values differ up to 21.9 % and for individual realizations this difference can be up to 59.4 %. We show via additional experiments that these differences in performance become negligible when using a folding horizon approach. The aim of this paper is to convince users of adjustable robust optimization to check for existence of multiple solutions. Using the production-inventory example and an illustrative toy example we deduce three important implications of the existence of multiple optimal robust solutions. First, if one neglects this existence of multiple solutions, then one can wrongly conclude that the adjustable robust solution does not outperform the nonadjustable robust solution. Second, even when it is a priori known that the adjustable and nonadjustable robust solutions are equivalent on worst-case objective value, they might still differ on the mean objective value. Third, even if it is known that affine decision rules yield (near) optimal performance in the adjustable robust optimization setting, then still nonlinear decision rules can yield much better mean objective values.
Original languageEnglish
Pages (from-to)531-545
JournalMathematical Programming
Volume160
Issue number1-2
Early online date30 Jan 2016
DOIs
Publication statusPublished - 11 Oct 2016

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Robust Optimization
Multiple Solutions
Decision Rules
Folding
Horizon
Deduce
Experiment
Experiments

Keywords

  • adjustable robust optimization
  • production-inventory problems
  • folding horizon
  • multiple solutions

Cite this

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title = "The impact of the existence of multiple adjustable robust solutions",
abstract = "In this note we show that multiple solutions exist for the production-inventory example in the seminal paper on adjustable robust optimization in Ben-Tal et al. (Math Program 99(2):351–376, 2004). All these optimal robust solutions have the same worst-case objective value, but the mean objective values differ up to 21.9 {\%} and for individual realizations this difference can be up to 59.4 {\%}. We show via additional experiments that these differences in performance become negligible when using a folding horizon approach. The aim of this paper is to convince users of adjustable robust optimization to check for existence of multiple solutions. Using the production-inventory example and an illustrative toy example we deduce three important implications of the existence of multiple optimal robust solutions. First, if one neglects this existence of multiple solutions, then one can wrongly conclude that the adjustable robust solution does not outperform the nonadjustable robust solution. Second, even when it is a priori known that the adjustable and nonadjustable robust solutions are equivalent on worst-case objective value, they might still differ on the mean objective value. Third, even if it is known that affine decision rules yield (near) optimal performance in the adjustable robust optimization setting, then still nonlinear decision rules can yield much better mean objective values.",
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The impact of the existence of multiple adjustable robust solutions. / de Ruiter, Frans; Brekelmans, Ruud; den Hertog, Dick.

In: Mathematical Programming , Vol. 160, No. 1-2, 11.10.2016, p. 531-545.

Research output: Contribution to journalArticleScientificpeer-review

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