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Keywords:
differentiability; delta-convex functions
differentiability; delta-convex functions
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.
References:
[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
[2] Kopecká E., Malý J.: Remarks on delta-convex functions. Comment. Math. Univ. Carolinae 31.3 (1990), 501-510. MR 1078484
[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. MR 0945661 | Zbl 0666.49005
[4] Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970). MR 0274683 | Zbl 0193.18401
[5] Shapiro A.: On functions representable as a difference of two convex functions in inequality constrained optimization. Research report University of South Africa, 1983.
[6] Veselý L., Zajíček L.: Delta-convex mappings between Banach spaces and applications. Dissertationes Math. 289 (1989), 1-52. MR 1016045
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