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Free Disposal Hull Condition to Verify When Efficiency Coincides with Weak Efficiency

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Abstract

In solving a multi-objective optimization problem by scalarization techniques, solutions to a scalarized problem are, in general, weakly efficient rather than efficient to the original problem. Thus, it is crucial to understand what condition ensures that all weakly efficient solutions are efficient. In this paper, we give a condition to verify when efficiency coincides with weak efficiency, provided that the free disposal hull of a given set is convex. By using this characterization, we obtain various applications to multi-objective optimization problems under some convex conditions. We also apply the main theorem to the least absolute shrinkage and selection operator (LASSO) and show that for a multi-objective version of LASSO, all weakly efficient solutions are efficient. Numerical simulation demonstrates that this equivalence is helpful to accelerate the hyper-parameter search for LASSO.

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Notes

  1. Although \(\theta ^*\) satisfying (7.4) may not be unique, for two arbitrary \(\theta ^*,{\tilde{\theta }}^*\) satisfying (7.4), we have \(X\theta ^*(w)=X{\tilde{\theta }}^*(w)\) for all \(w\in [0,1]\) [35, Lemma 1(ii)].

  2. It aligns to the fact that R-package glmnet examines 100 hyper-parameters by default.

  3. https://www.cc.kyushu-u.ac.jp/scp/eng/system/ITO/01_intro.html.

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Acknowledgements

The authors are grateful to the reviewers for their comments and suggestions. We are also grateful to Kenta Hayano, Yutaro Kabata, and Hiroshi Teramoto for their kind comments. Shunsuke Ichiki was supported by JSPS KAKENHI Grant Numbers JP21K13786, JP19J00650 and JP17H06128. This work is based on the discussions at 2018 IMI Joint Use Research Program, Short-term Joint Research “multi-objective optimization and singularity theory: Classification of Pareto point singularities” in Kyushu University.

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Correspondence to Naoki Hamada.

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Communicated by Alfredo N. Iusem.

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Hamada, N., Ichiki, S. Free Disposal Hull Condition to Verify When Efficiency Coincides with Weak Efficiency. J Optim Theory Appl 192, 248–270 (2022). https://doi.org/10.1007/s10957-021-01961-5

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