Local boundedness and continuity of generalized convex functions

Hartwig, Helga

doi:10.1080/02331939208843838

Cited by 17 publications

(6 citation statements)

References 14 publications

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“…3 says that a piece of cheese is roughly convex, i.e., outer -convex w.r.t. = B(0, ), provided that the diameters of cheese holes are less than .…”

Section: Let X Be a Normed Vector Space C ⊂ X Be Convex S ⊂ X And Dmentioning

confidence: 99%

“…Other kinds of roughly convex functions, such as global -convexity introduced by Hu et al [2], rough -convexity proposed by Klötzler and investigated by Hartwig [3], -convexity defined in Phu [4,5], and symmetrical -convexity in Hai and Phu [6], are special cases of the outer -convexity studied in Phu and An [1], hence they do belong to the outer -convexity, too. One of the main aims for the mentioned rough generalizations is to get a wide class of nonconvex functions that are somehow roughly convex in order to still possess some similar optimization properties as of convex functions, e.g., lower level sets of roughly convex functions are roughly convex and local minimizers of roughly convex functions are global minimizers, where the notion of local minimizers must be modified accordingly.…”

mentioning

confidence: 97%

“…Almost all known kinds of roughly convex functions, such as globalconvexity in Hu et al [2], rough -convexity in Hartwig [3], -convexity in Phu [4,5], symmetrical -convexity in Hai and Phu [6], and outerconvexity in Phu and An [1], do belong to the outer -convexity. Hence, many examples of outer -convex functions can be read there and also in Phu [17].…”

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confidence: 99%

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Outer Γ-Convexity in Vector Spaces

Phú

2008

Numerical Functional Analysis and Optimization

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“…3 says that a piece of cheese is roughly convex, i.e., outer -convex w.r.t. = B(0, ), provided that the diameters of cheese holes are less than .…”

Section: Let X Be a Normed Vector Space C ⊂ X Be Convex S ⊂ X And Dmentioning

confidence: 99%

mentioning

confidence: 97%

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Outer Γ-Convexity in Vector Spaces

Phú

2008

Numerical Functional Analysis and Optimization

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“…There are now three classes of roughly convex functions: -8-convexity: &convex, midpoint &convex, 6-quasiconvex, and midpoint 6-quasiconvex functions, which were introduced by Hu, Klee, and Larman in [5]; -y-convexity: y-convex, y-quasiconvex, lightly y-convex, and midpoint y-convex functions, which were introduced by Phu in [lo, 111, and [14]; -p-convex functions, which were introduced by Klotzler, Hartwig, and Sollner in [4] and [I81 (actually, they were called "roughly r-convex", but we would like to take this name over for all generalizations in this direction).…”

Section: Introductionmentioning

confidence: 99%

“…Hartwig [4] and Sollner [I81 concerned with the question when and where a p-convex function is bounded or continuous.…”

Section: Introductionmentioning

confidence: 99%

Some properties of globally δ-convex functions^∗

Phú

1995

Optimization

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The notions of d-convex and midpoint 8-convex functions were introduced by Hu, Klee, and Larman (SIAM Journal on Control and Optimization, Vol. 27, 1989). It is known that such functions have some important optimization properties: each r-local minimum is a global minimum, and if they assume their global maximum on a bounded convex domain of a Hilbert space then they do so at least at some r-extreme points of this domain. In this paper, some analytical properties of 6-convex and midpoint d-convex functions are investigated. Concretely, it is shown when they are bounded (from above or from below). For instance, a 6-convex function defined on the entire real line is always locally bounded, and midpoint d-convex function on the real line is either locally bounded or totally unbounded. Further on, it is proved that there are totally discontinuous (i.e., nowhere differentiable) &convex and midpoint 6-convex functions on the real line.

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Are Generalized Derivatives Sseful for Generalized Convex Functions?

Penot

1998

Nonconvex Optimization and Its Applications

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We present a review of some ad hoc sub differentials which have been devised for the needs of generalized convexity such as the quasi-sub differentials of GreenbergPierskalla, the tangential of Crouzeix, the lower sub differential of Plastria, the infradifferential of Gutierrez, the subdifferentials of Martinez-Legaz-Sach, Penot-Volle, Thach. We complete this list by some new proposals. We compare these specific subdifferentials to some all-purpose sub differentials used in nonsmooth analysis. We give some hints about their uses. We also point out links with duality theories.

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Local boundedness and continuity of generalized convex functions

Cited by 17 publications

References 14 publications

Outer Γ-Convexity in Vector Spaces

Outer Γ-Convexity in Vector Spaces

Some properties of globally δ-convex functions^∗

Are Generalized Derivatives Sseful for Generalized Convex Functions?

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Local boundedness and continuity of generalized convex functions

Cited by 17 publications

References 14 publications

Outer Γ-Convexity in Vector Spaces

Outer Γ-Convexity in Vector Spaces

Some properties of globally δ-convex functions∗

Are Generalized Derivatives Sseful for Generalized Convex Functions?

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Product

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Some properties of globally δ-convex functions^∗