A d.c. function need not be difference of convex functions
Commentationes Mathematicae Universitatis Carolinae (2005)
- Volume: 46, Issue: 1, page 75-83
- ISSN: 0010-2628
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topPavlica, David. "A d.c. $C^1$ function need not be difference of convex $C^1$ functions." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 75-83. <http://eudml.org/doc/249529>.
@article{Pavlica2005,
abstract = {In [2] a delta convex function on $\mathbb \{R\}^2$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class $C^1(\mathbb \{R\}^2)$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.},
author = {Pavlica, David},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {differentiability; delta-convex functions; diferentiability; delta-convex functions},
language = {eng},
number = {1},
pages = {75-83},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A d.c. $C^1$ function need not be difference of convex $C^1$ functions},
url = {http://eudml.org/doc/249529},
volume = {46},
year = {2005},
}
TY - JOUR
AU - Pavlica, David
TI - A d.c. $C^1$ function need not be difference of convex $C^1$ functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 75
EP - 83
AB - In [2] a delta convex function on $\mathbb {R}^2$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class $C^1(\mathbb {R}^2)$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.
LA - eng
KW - differentiability; delta-convex functions; diferentiability; delta-convex functions
UR - http://eudml.org/doc/249529
ER -
References
top- Hiriart-Urruty J.-B., Generalized differentiability, duality and optimization for problem dealing with difference of convex functions, Lecture Notes in Econom. and Math. Systems 256 J. Ponstein, Ed., Springer, Berlin, 1985, pp. 37-70. MR0873269
- Kopecká E., Malý J., Remarks on delta-convex functions, Comment. Math. Univ. Carolinae 31.3 (1990), 501-510. (1990) MR1078484
- Penot J.-P., Bougeard M.L., Approximation and decomposition properties of some classes of locally d.c. functions, Math. Programming 41 (1988), 195-227. (1988) Zbl0666.49005MR0945661
- Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton (1970). (1970) Zbl0193.18401MR0274683
- Shapiro A., On functions representable as a difference of two convex functions in inequality constrained optimization, Research report University of South Africa, 1983.
- Veselý L., Zajíček L., Delta-convex mappings between Banach spaces and applications, Dissertationes Math. 289 (1989), 1-52. (1989) MR1016045
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