Convexity and Well-Posed Problems

Lucchetti, Roberto

doi:10.1007/0-387-31082-7

Cited by 175 publications

(112 citation statements)

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“…Therefore, (12) yields that (0 n ; 1) 2 cl K ( " ) and " = 2 dom F b ; a contradiction. This shows that (ii) =) (v) and (iii) =) (v); and the result follows.…”

Section: Perturbing the Left-hand Side Datamentioning

confidence: 95%

“…The (positive) polar cone of a convex cone X is X := fy 2 R n : x 0 y 0; x 2 Xg. For more on these concepts and in general on convexity, see [6,12].…”

Section: Remark 11 (Lower Semicontinuity Along Lines) It Follows Rementioning

confidence: 99%

“…In a metric setting this can alternatively be expressed by saying that M(y) Liminf M(y k ) for every sequence fy k g Y such that y k ! y, where Liminf A k denotes the Kuratowski-Painlevé inner limit of a sequence of the closed sets A k (see [12], [14]). We also say that a point x 2 M(y) is lower stable for M at y if for every open set O 3 x there is an open set U containing y such that M(z) \ O 6 = ; for all z 2 U , that is, x 2 Liminf M(y k ) for every sequence fy k g Y converging to y.…”

Section: Introductionmentioning

confidence: 99%

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Lower Semicontinuity of the Feasible Set Mapping of Linear Systems Relative to Their Domains

Daniilidis

Goberna

López

et al. 2012

Set-Valued Var. Anal

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Abstract. This paper deals with stability properties of the feasible set of linear inequality systems having a …nite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, a¤ecting respectively, all of the data, the left-hand side data, or the right-hand side coe¢ cients. Mathematics Subject Classi…cation (2000):Primary: 49K40, 90C34; Secondary: 15A39, 90C05.

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“…Therefore, (12) yields that (0 n ; 1) 2 cl K ( " ) and " = 2 dom F b ; a contradiction. This shows that (ii) =) (v) and (iii) =) (v); and the result follows.…”

Section: Perturbing the Left-hand Side Datamentioning

confidence: 95%

“…The (positive) polar cone of a convex cone X is X := fy 2 R n : x 0 y 0; x 2 Xg. For more on these concepts and in general on convexity, see [6,12].…”

Section: Remark 11 (Lower Semicontinuity Along Lines) It Follows Rementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lower Semicontinuity of the Feasible Set Mapping of Linear Systems Relative to Their Domains

Daniilidis

Goberna

López

et al. 2012

Set-Valued Var. Anal

Self Cite

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“…Furthermore, since Γ-convergence is tied to Convex Analysis [2,3,4,5,6,8,21,24], we study some questions of convergence in the supremum metric from this point of view. Specifically, we analyze how this convergence behaves with respect to maximizers.…”

Section: Introductionmentioning

confidence: 99%

Convergence of fuzzy sets with respect to the supremum metric

Pedraza

López

Romaguera

2014

Fuzzy Sets and Systems

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ElsevierPedraza Aguilera, T.; Rodríguez López, J.; Romaguera Bonilla, S. (2014). Convergence of fuzzy sets with respect to the supremum metric. Fuzzy Sets and Systems. 245:83-100. doi:10.1016/j.fss.2014.03.005. CONVERGENCE OF FUZZY SETS WITH RESPECT TO THE SUPREMUM METRICTATIANA PEDRAZA, JESÚS RODRÍGUEZ-LÓPEZ ⋆, * AND SALVADOR ROMAGUERA * Abstract. We characterize the convergence of fuzzy sets in the supremum metric given by the supremum of the Hausdorff distances of the α-cuts of the fuzzy sets. We do it by dividing this metric into its lower and upper quasipseudometric parts. This characterization is given in the more general context with no assumption on the fuzzy sets. Furthermore, motivated from the theory of Convex Analysis, we also provide some results about the behaviour of the convergence in the supremum metric with respect to maximizers.

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“…so that the topology induced by on is the product of the topologies induced by the Hausdor¤ metric (5) on 1 and the Hausdor¤-like metric (3) on 2 (the induced topology on 2 coincides with the so-called bounded Hausdor¤ topology; see, e.g., [14]), respectively. Our feasible set mapping is F :…”

Section: Introductionmentioning

confidence: 99%