Sets of efficient points in a normed space

Durier, Roland; Michelot, Christian

doi:10.1016/0022-247x(86)90237-4

Cited by 48 publications

(37 citation statements)

References 7 publications

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“…The idea of a staircase operator is closely related to the so-called "diff-max" property of convex functions [8,9,29]. In brief, a convex function h is diff-max if and only if (Oh) -1 is staircase.…”

Section: Concerning Finite Terminationmentioning

confidence: 99%

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Eckstein¹,

Bertsekas

1992

Mathematical Programming

2,480

2,027

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This paper shows, by means of an operator called a splitting operator, that the Douglas-Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm, Therefore, applications of Douglas-Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new, generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.

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Section: Concerning Finite Terminationmentioning

confidence: 99%

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Eckstein¹,

Bertsekas

1992

Mathematical Programming

2,480

2,027

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“…Now we are in the right position to show the main result about the geometrical structure of X w Par .f 1 ; : : : ; f Q /. -Taking f i .x/ D kx a i k with a i 2 R 2 for i D 1; : : : ; Q and k k being a strictly convex norm or a norm derived from a scalar product, we get Proposition 1.3, Theorem 4.3 and Corollary 4.1 in Durier and Michelot (1986). The set of weakly efficient locations is the convex hull of the points a i with i D 1; : : : ; Q.…”

Section: -Facility Planar/continuous Location Problemsmentioning

confidence: 97%

Location Problems with Multiple Criteria

2015

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This chapter analyzes multicriteria continuous, network, and discrete location problems. In the continuous framework, we provide a complete description of the set of weak Pareto, Pareto, and strict Pareto locations for a general Q-criteria location problem based on the characterization of three criteria problems. In the network case, the set of Pareto locations is characterized for general networks as well as for tree networks using the concavity and convexity properties of the distance function on the edges. In the discrete setting, the entire set of Pareto locations is characterized using rational generating functions of integer points in polytopes. Moreover, we describe algorithms to obtain the solutions sets (the different Pareto locations) using the above characterizations. We also include a detailed complexity analysis. A number of references has been cited throughout the chapter to avoid the inclusion of unnecessary technical details and also to be useful for a deeper analysis.

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“…Dans [7], nous avons étudié le problème classique de localisation multicritère dans un espace réel norme N, muni de la norme v, pour un ensemble nomaécessairement fini de points^ de N, les normes associées aux différents points a étant toutes égales à v. Le point de vue géométrique adopté nous a conduit à introduire des ensembles notés Q h dans la publication citée.…”

Section: *1 Cônes Q ô Dans Un Espace Normeunclassified

“…Remarque 6 : Si les sous-espaces X i sont tous de dimension 1 (7 norme vectorielle de taille n) 7 alors y est nécessairement polyédrique. On retrouve ainsi la convexité-compacité de S dans le cadre de la Proposition 3.…”

Section: Un Résultat Préliminaireunclassified

Meilleure approximation en norme vectorielle et théorie de la localisation

Durier¹

1987

ESAIM: M2AN

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Sets of efficient points in a normed space

Cited by 48 publications

References 7 publications

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

Location Problems with Multiple Criteria

Meilleure approximation en norme vectorielle et théorie de la localisation

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