| Title:
|
A d.c. $C^1$ function need not be difference of convex $C^1$ functions (English) |
| Author:
|
Pavlica, David |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
1 |
| Year:
|
2005 |
| Pages:
|
75-83 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In [2] a delta convex function on $\Bbb R^2$ 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 $C^1(\Bbb R^2)$ 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. (English) |
| Keyword:
|
differentiability |
| Keyword:
|
delta-convex functions |
| MSC:
|
26B05 |
| MSC:
|
26B25 |
| idZBL:
|
Zbl 1121.26011 |
| idMR:
|
MR2175860 |
| . |
| Date available:
|
2009-05-05T16:49:41Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119509 |
| . |
| Reference:
|
[1] 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. MR 0873269 |
| Reference:
|
[2] Kopecká E., Malý J.: Remarks on delta-convex functions.Comment. Math. Univ. Carolinae 31.3 (1990), 501-510. MR 1078484 |
| Reference:
|
[3] Penot J.-P., Bougeard M.L.: Approximation and decomposition properties of some classes of locally d.c. functions.Math. Programming 41 (1988), 195-227. Zbl 0666.49005, MR 0945661 |
| Reference:
|
[4] Rockafellar R.T.: Convex Analysis.Princeton University Press, Princeton (1970). Zbl 0193.18401, MR 0274683 |
| Reference:
|
[5] Shapiro A.: On functions representable as a difference of two convex functions in inequality constrained optimization.Research report University of South Africa, 1983. |
| Reference:
|
[6] Veselý L., Zajíček L.: Delta-convex mappings between Banach spaces and applications.Dissertationes Math. 289 (1989), 1-52. MR 1016045 |
| . |
