1. CMB 2005 (vol 48 pp. 283)
|Enlarged Inclusion of Subdifferentials |
This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $ \gamma$-Lipschitz over their effective domain.
Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable function
Categories:49J52, 46N10, 58C20
2. CMB 2000 (vol 43 pp. 25)
|Subdifferential Regularity of Directionally Lipschitzian Functions |
Formulas for the Clarke subdifferential are always expressed in the form of inclusion. The equality form in these formulas generally requires the functions to be directionally regular. This paper studies the directional regularity of the general class of extended-real-valued functions that are directionally Lipschitzian. Connections with the concept of subdifferential regularity are also established.
Keywords:subdifferential regularity, directional regularity, directionally Lipschitzian functions
Categories:49J52, 58C20, 49J50, 90C26
3. CMB 1998 (vol 41 pp. 497)
|On the construction of HÃ¶lder and Proximal Subderivatives |
We construct Lipschitz functions such that for all $s>0$ they are $s$-H\"older, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed H\"older and approximate subderivatives.
Keywords:Lipschitz functions, HÃ¶lder subdifferential, proximal subdifferential, approximate subdifferential, symmetric subdifferential, HÃ¶lder smooth, dyadic rationals
Categories:49J52, 26A16, 26A24