Generic Existence of Solutions for Some Perturbed Optimization Problems

Cobzaş, Ştefan

doi:10.1006/jmaa.1999.6675

Cited by 21 publications

(9 citation statements)

References 22 publications

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“…The following corollary is the main result in [5], it extends a result of Lau [11] on nearest points (the case J ≡ 0). [5,Theorem 2.1].…”

Section: Then F Z (·) Is a Lipschitz Function On Q And So By Lemma 2supporting

confidence: 76%

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Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Ren-xing

2005

Journal of Mathematical Analysis and Applications

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Let Z be a closed, boundedly relatively weakly compact, nonempty subset of a Banach space X, and J : Z → R a lower semicontinuous function bounded from below. If X 0 is a convex subset in X and X 0 has approximatively Z-property (K), then the set of all points x in X 0 \ Z for which there exists z 0 ∈ Z such that J (z 0 ) + x − z 0 = ϕ(x) and every sequence {z n } ⊂ Z satisfying lim n→∞ [J (z n ) + x − z n ] = ϕ(x) for x contains a subsequence strongly convergent to an element of Z is a dense G δ -subset of X 0 \ Z. Moreover, under the assumption that X 0 is approximatively Z-strictly convex, we show more, namely that the set of all points x in X 0 \ Z for which there exists a unique point z 0 ∈ Z such that J (z 0 ) + x − z 0 = ϕ(x) and every sequence {z n } ⊂ Z satisfying lim n→∞ [J (z n ) + x − z n = ϕ(x) for x converges strongly to z 0 is a dense G δ -subset of X 0 \ Z. Here ϕ(x) = inf{J (z) + x − z ; z ∈ Z}. These extend S. Cobzas's result [

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“…The following corollary is the main result in [5], it extends a result of Lau [11] on nearest points (the case J ≡ 0). [5,Theorem 2.1].…”

Section: Then F Z (·) Is a Lipschitz Function On Q And So By Lemma 2supporting

confidence: 76%

“…Recently, S. Cobzas [5] extends Baranger's results [1] mentioned above to reflexive Kadec Banach space. For other results on perturbed optimization problems, see [9,10,12,15] and the monograph [8].…”

Section: Introductionmentioning

confidence: 70%

Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Ren-xing

2005

Journal of Mathematical Analysis and Applications

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“…For more developments and extensions in this direction, the readers are referred to [2,3,7,8,11,[22][23][24][25][26][27][28][29][31][32][33] and the surveys [12,30].…”

Section: Introductionmentioning

confidence: 99%

Nearest and farthest points in spaces of curvature bounded below

Espínola

López-Acedo

2010

Journal of Approximation Theory

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Let A be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space (X, d) and x ∈ X \ A. The nearest point problem (resp. the farthest point problem) w.r.t. x considered here is to find a point a 0 ∈ A such that d(x, a 0 ) = inf{d(x, a) : a ∈ A} (resp. d(x, a 0 ) = sup{d(x, a) : a ∈ A}). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if X is a space of curvature bounded below then the complement of the set of all points x ∈ X for which the nearest point problem (resp. the farthest point problem) w.r.t. x is well posed is σ -porous in X \ A. Our results extend and/or improve some recent results about proximinality in geodesic spaces as well as the corresponding ones previously obtained in uniformly convex Banach spaces. In particular, the result regarding the nearest point problem answers affirmatively an open problem proposed by Kaewcharoen and Kirk [A. Kaewcharoen, W.A. Kirk, Proximinality in geodesic spaces, Abstr.

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“…The existence results have been applied to optimal control problems governed by partial differential equations, see, for example, [3,4,6,[8][9][10]14,24]. Here we are especially interested in the study of the problem (x, J )-sup; while, for the problem (x, J )-inf, the readers are referred to [6,10,11,20,26]. In [5], it was proved that if X is a reflexive and locally uniformly convex Banach space then the set of all x ∈ X for which the problem (x, J )-sup has a solution is a dense G δ -subset of X.…”

Section: Introductionmentioning

confidence: 99%

Existence and porosity for a class of perturbed optimization problems in Banach spaces

2007

Journal of Mathematical Analysis and Applications

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Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup z∈Z {J (z) + x − z }, which is denoted by (x, J )-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈ X for which the problem (x, J )-sup has a solution is a dense G δ -subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z 0 ∈ Z such that J (z 0 ) + x − z 0 = sup z∈Z {J (z) + x − z } is a σ -porous subset of X and the set of all points x ∈ X \ Z 0 such that there exists a maximizing sequence of the problem (x, J )-sup which has no convergent subsequence is a σ -porous subset of X \ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z is in the solution set of (z, J )-sup.

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Generic Existence of Solutions for Some Perturbed Optimization Problems

Abstract: Let X be a real Banach space, Z a closed nonvoid subset of X, and J: Z ª ‫ޒ‬ a lower semicontinuous function bounded from below. If X is reflexive and has the Kadets property then the set of all x g X for which there exists z g Z such that

Cited by 21 publications

References 22 publications

Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Nearest and farthest points in spaces of curvature bounded below

Existence and porosity for a class of perturbed optimization problems in Banach spaces

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