D.C. Representability of closed sets in reflexive Banach spaces and applications to optimization problems

Thach, Phan Thien; Konno, Hiroshi

doi:10.1007/bf02192279

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Functions and Nearest Points3

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2015

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(3 citation statements)

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“…In particular, every DC representable set in X is a projection of a DC set in X ⊕ R. The next theorem was proved in [32].…”

Section: Functions and Nearest Pointsmentioning

confidence: 91%

“…In particular, every DC representable set in X is a projection of a DC set in X ⊕ R. The following theorem was proved in [TK96] Theorem 2.3 (Thach-Konno [TK96]). If X is a reflexive Banach space and C ⊆ X is closed, then C is DC representable.…”

Section: Functions and Nearest Pointsmentioning

confidence: 98%

“…Theorem 2.12 [32]. If X is a reflexive Banach space and C ⊆ X is closed, then C is DC representable.…”

Section: Functions and Nearest Pointsmentioning

confidence: 99%

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Nearest Points and Delta Convex Functions in Banach Spaces

Borwein¹,

Giladi²

2015

Bull. Aust. Math. Soc.

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Abstract. Given a closed set C in a Banach space (X, · ), a point x ∈ X is said to have a nearest point in C if there exists z ∈ C such that d C (x) = x − z , where d C is the distance of x from C. We shortly survey the problem of studying how large is the set of points in X which have nearest points in C. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.

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“…In particular, every DC representable set in X is a projection of a DC set in X ⊕ R. The next theorem was proved in [32].…”

Section: Functions and Nearest Pointsmentioning

confidence: 91%

Section: Functions and Nearest Pointsmentioning

confidence: 98%