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Semivectorial Bilevel Optimization on Riemannian Manifolds

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Abstract

In this paper, we deal with the semivectorial bilevel problem in the Riemannian setting. The upper level is a scalar optimization problem to be solved by the leader, and the lower level is a multiobjective optimization problem to be solved by several followers acting in a cooperative way inside the greatest coalition and choosing among Pareto solutions with respect to a given ordering cone. For the so-called optimistic problem, when the followers choice among their best responses is the most favorable for the leader, we give optimality conditions. Also for the so-called pessimistic problem, when there is no cooperation between the leader and the followers, and the followers choice may be the worst for the leader, we present an existence result.

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Notes

  1. We use the notations “MIN” and “ARGMIN” for vector-valued functions to distinguish from the notations “min” and “argmin” used for scalar-valued functions.

  2. This fact is not true in general, i.e., when C is a cone in a topological vector space, but in our setting, we take advantage of the finite dimension of \(\mathbb {R}^r\).

  3. This hypothesis holds, for example, if we assume that G is a conformal map, or, for example, if there exists a real number \(c > 0\) such that \(g_2 (\delta _2 G (\lambda _0 , x_0 , y_0) (v),v) \ge c g_2 (v,v),\;\forall v \in T_yM_2\).

  4. i.e., there exists an open neighborhood \(\mathcal {N}=\mathcal {N}_1\times \mathcal {N}'\subset M_1\times \Lambda _p\) of \((x^*,\lambda ^*)\) such that, for each \(x\in \mathcal {N}_1\), the function \(\mathcal {N}'\ni \lambda \mapsto f(x,y(x,\lambda ))\) admits local minimizers, and for all \(x\in \mathcal {N}_1\), \(f(x^*,\lambda ^*) \le \min _{\lambda \in \mathcal {N}'}f(x,y(x,\lambda ))\).

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Acknowledgments

The authors are very grateful to the anonymous referees for their useful comments and suggestions which have improved the quality of the paper.

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Authors and Affiliations

  1. ERIM, Université de la Nouvelle-Calédonie, BP. R4, 98851, Nouméa Cédex, New Caledonia, France

    Henri Bonnel

  2. Institut de Mathématiques et de Sciences Physiques, Porto-Novo, Benin

    Léonard Todjihoundé

  3. Department of Mathematics-Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Splaiul Independentei 313, 060042, Bucharest, Romania

    Constantin Udrişte

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  1. Henri Bonnel
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Correspondence to Henri Bonnel.

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Communicated by Johannes Jahn.

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Bonnel, H., Todjihoundé, L. & Udrişte, C. Semivectorial Bilevel Optimization on Riemannian Manifolds. J Optim Theory Appl 167, 464–486 (2015). https://doi.org/10.1007/s10957-015-0789-6

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