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Optimizing over the properly efficient set of convex multi-objective optimization problems

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Abstract

Optimizing over the efficient set of a multi-objective optimization problem is among the difficult problems in global optimization because of its nonconvexity, even in the linear case. In this paper, we consider only properly efficient solutions which are characterized through weighted sum scalarization. We propose a numerical method to tackle this problem when the objective functions and the feasible set of the multi-objective optimization problem are convex. This algorithm penalizes progressively iterates that are not properly efficient and uses a sequence of convex nonlinear subproblems that can be solved efficiently. The proposed algorithm is shown to perform well on a set of standard problems from the literature, as it allows to obtain optimal solutions in all cases.

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Notes

  1. In particular, we have observed this situation for problem \(P_9\) when \(\beta \ge 100\). The reason is that \(\gamma \) increases too fast hence \(\phi (x)\) is not given enough importance in the objective function.

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Acknowledgements

The first and third authors acknowledge the support of the Direction Générale de la Recherche Scientifique et du Développement Technologique (DGRSDT) Grant ID: C0656104. We are grateful to the reviewer whose comments allowed us to improve the manuscript significantly.

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Authors and Affiliations

  1. LaROMad, Faculty of Mathematics (USTHB), Algiers, Algeria

    Kahina Ghazli & Mustapha Moulaï

  2. University of Bejaia, Béjaïa, Algeria

    Kahina Ghazli

  3. Department of Mathematics and Operational Research, Faculté Polytechnique, University of Mons, Mons, Belgium

    Nicolas Gillis

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  1. Kahina Ghazli
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Correspondence to Kahina Ghazli.

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Ghazli, K., Gillis, N. & Moulaï, M. Optimizing over the properly efficient set of convex multi-objective optimization problems. Ann Oper Res 295, 575–604 (2020). https://doi.org/10.1007/s10479-020-03820-4

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