The set of all nondominated solutions in linear cases and a multicriteria simplex method

Yu, Po-Lung; Zelený, Milan

doi:10.1016/0022-247x(75)90189-4

Cited by 360 publications

(128 citation statements)

References 6 publications

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“…We will later uae MCLP to define connectednese in discrete multiple criteria optimization. Before the general result of Theorem 1 was known the connectedness result for MCLP had been proved by varioua authora [2,4,12]. The most important solutions in linear programming are basic solutions which correspond to vertices of the polyhedral feasible set X, and fundamental solutions which correspond to extreme rays of X, if X is unbounded.…”

mentioning

confidence: 99%

Connectedness of efficient solutions in multiple criteria combinatorial optimization

Ehrgott

Klamroth

1997

European Journal of Operational Research

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mentioning

confidence: 99%

Connectedness of efficient solutions in multiple criteria combinatorial optimization

Ehrgott

Klamroth

1997

European Journal of Operational Research

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“…Convex multiobjective optimization problems have been studied extensively (see e.g., Ruzika and Wiecek 2005). For our treatment of this case, we need a lemma from (Yu and Zeleny 1975). …”

Section: Convex Casementioning

confidence: 99%

Constructing a Pareto front approximation for decision making

2011

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An approach to constructing a Pareto front approximation to computationally expensive multiobjective optimization problems is developed. The approximation is constructed as a sub-complex of a Delaunay triangulation of a finite set of Pareto optimal outcomes to the problem. The approach is based on the concept of inherent nondominance. Rules for checking the inherent nondominance of complexes are developed and applying the rules is demonstrated with examples. The quality of the approximation is quantified with error estimates. Due to its properties, the Pareto front approximation works as a surrogate to the original problem for decision making with interactive methods.

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“…When X is a polyhedron and P is a polyhedral cone, parametric analysis in linear programming shows that A(X) is connected (see, for example, Koopmans [11], or Yu and Zeleny [30] for related results). To study the connectedness of A(X) in a more general setting, we also use a parametrization of a mathematical programming objective function.…”

Section: Connectednessmentioning

confidence: 99%

“…and Kouada [23], [24], Evans and Steuer [25], Gal [26], Geoffrion, Dyer and Feinberg [27], Philip [28], Sachtman 29], and Yu and Zeleny [30]) have treated algorithms for determining and investigating admissible points, primarily for linear problems. '…”

Section: Introductionmentioning

confidence: 99%

The structure of admissible points with respect to cone dominance

Bitran

Magnanti

1979

J Optim Theory Appl

122

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We study the set of admissible (pareto-optimal) points of a closed convex set X when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions of X, we (i) extend a representation theorem of Arrow, Barankin and Blackwell by showing that all admissible points are either limit points of certain "strictly admissible" alternatives or translations of such limit points by rays in the closure of the preference cone, and (ii) show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed upon X, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element of X to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.

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The set of all nondominated solutions in linear cases and a multicriteria simplex method

Cited by 360 publications

References 6 publications

Connectedness of efficient solutions in multiple criteria combinatorial optimization

Connectedness of efficient solutions in multiple criteria combinatorial optimization

Constructing a Pareto front approximation for decision making

The structure of admissible points with respect to cone dominance

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