Characterizations of metric projections in Banach spaces and applications

Penot, Jean-Paul; Ratsimahalo, Robert

doi:10.1155/s1085337598000451

Cited by 17 publications

(9 citation statements)

References 23 publications

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“…The metric projection can be characterized via the duality maps. This result was established in Penot and Ratsimahalo [17]. For the convenience of the reader, we will present a proof below.…”

Section: Generalized Normal Maps and Their Localizationmentioning

confidence: 57%

Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

Kien

Yao

2007

Set-Valued Anal

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In this paper, we introduce a localized version of generalized normal maps as well as generalized natural mappings. By using these concepts, we study continuity properties of the solution map of parametric variational inequalities in reflexive Banach spaces. This localization permits us to deal with variational conditions posed on sets that may not be convex and to establish existence and continuity of solutions. We also establish homeomorhisms between the solution set of variational inequalities and the solution set of generalized normal maps. Using these homeomorphisms and the degree theory, we show that the solution map of parametric variational inequalities is lower semicontinuous. Our results extend some results of Robinson (Set-Valued Anal 12: [259][260][261][262][263][264][265][266][267][268][269][270][271][272][273][274] 2004).

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Section: Generalized Normal Maps and Their Localizationmentioning

confidence: 57%

Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

Kien

Yao

2007

Set-Valued Anal

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“…Note that (27) and (29) coincide with (21) and (22), respectively. Since i ¼ ÀG i (x*) by (28), we get (23) and (24) from (30) and (31), respectively.…”

Section: Proofmentioning

confidence: 69%

“…which in turn implies that À1 i j i ¼ ÀP ÀC i ðM r i ðx j , " j , j ÞÞ (see Theorem 3.1 in [27]). Consequently, (35)-(38) can be rewritten as…”

Section: Theorem 44mentioning

confidence: 93%

A proximal augmented Lagrangian method for equilibrium problems

Mashreghi

Nasri

2012

Applicable Analysis

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Considering a recently proposed proximal point method for equilibrium problems, we construct an augmented Lagrangian method for solving the same problem in reflexive Banach spaces with cone constraints generating a strongly convergent sequence to a certain solution of the problem. This is an inexact hybrid method meaning that at a certain iterate, a solution of an unconstrained equilibrium problem is found, allowing a proper error bound, followed by a Bregman projection of the initial iterate onto the intersection of two appropriate halfspaces. Assuming a set of reasonable hypotheses, we provide a full convergence analysis.

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“…Definition 3.4 is different from Definition 2.8 due to technical details about metric projections in Banach spaces discussed by Penot [40]. However, their essence is the same and they are equivalent in reflexive Banach spaces.…”

Section: The Connection Lemma In Banach Spacesmentioning

confidence: 99%