Dualität und algorithmische anwendung beim vektoriellen standortproblem

Gerth, Christiane; Pöhler, Karin

doi:10.1080/02331938808843369

Cited by 15 publications

(6 citation statements)

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“…The following result from [50] is a counterpart of the characterization of solutions x ∈ R 2 with G(x) ∈ Eff(G(R 2 ); R q + ) (see Remark 3.1) proposed by the authors in [53], where multiobjective location problems defined by the maximum-norm are considered. Proposition 5.1.…”

Section: 31mentioning

confidence: 99%

“…The following preparation allows us to characterize the set χ Eff (G(R 2 ); R q + ) of efficient solutions of the multi-objective location problem (MOLP). In the literature, several characterizations of the set efficient solutions of location problems have been obtained; see, e.g., [50,52,53]. We will adapt here the approach proposed by the authors in [53] which is based on the dual norm to the Manhattan-norm, namely the maximum-norm defined for any (x 1 , x 2 ) ∈ R 2 by…”

Section: 31mentioning

confidence: 99%

“…The authors in [53] characterized the set of efficient solutions of multi-objective location problems, where the distances are given by the Manhattan-norm, using the smallest sublevel set of the dual norm to the Manhattan-norm (i.e., the maximum-norm) containing the points a i ∈ R 2 , i = 1, 2, . .…”

Section: 31mentioning

confidence: 99%

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Existence results and optimization over the set of efficient solutions in vector-valued approximation theory

2023

J. Nonlinear Var. Anal.

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Continuity of the objective functions and compactness of their domain are classical assumptions widely used to obtain existence results for solutions of optimization problems. Due to the lack of compactness in general spaces, under some moderate assumptions concerning the objective function and the feasible set, we derive existence results for vector-valued optimization problems and corresponding results for associated scalarized problems in this paper. Furthermore, we apply our results to special vector-valued approximation problems, especially to multi-objective location problems where the whole set of efficient solutions can be generated by a geometric primal-dual algorithm. Moreover, by using the nonlinear scalarizing functional introduced by Gerstewitz, we perform an optimization according to the preferences of a decision maker on the generated set of efficient solutions from which we derive a single solution of this set that corresponds to the preferences of the decision maker.

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Section: 31mentioning

confidence: 99%

Section: 31mentioning

confidence: 99%

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Existence results and optimization over the set of efficient solutions in vector-valued approximation theory

2023

J. Nonlinear Var. Anal.

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“…those by GERTH, POHLER [5], GERTH, GOPFERT, POHLER [6] and WANKA [9], [lo], the present paper considers contributions concerned with the duality of such problems involving additional restrictions in the form of inequalities. We formu!ate a dual vecterial maximl-rm prohiem and weak duality for a glven general vzctorial minimum problem and a certain SirGng duality reiation for properly efifrc-ient solutions.…”

mentioning

confidence: 99%

“…We formu!ate a dual vecterial maximl-rm prohiem and weak duality for a glven general vzctorial minimum problem and a certain SirGng duality reiation for properly efifrc-ient solutions. I n GERTI:, P~H L E R , G~P F E R T [5], [6], vectorial location problems in Rn without restrictions are considered. In papers -A -.…”

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confidence: 99%