Topological characterization of the approximate subdifferential
We study topological properties of the subdifferential introduced by Mordukhovich (also called the approximate subdifferential {partial_derivative}{sub a}). In contrast to other types of subdifferential, {partial_derivative}{sub a}f(x) is not convex in general. It is demonstrated that quite a large variety of topological types may occur. In a restricted infinite dimensional setting (separable Hilbert space H) one may verify the following approximation result for {partial_derivative}{sub a} (according to the extended definition by loffe): Given any weakly compact subset K {improper_subset} H there exists a sequence of lipschitzian functions f{sub n} : H {times} R {yields} R such that lim{sub n{yields}{infinity}}{partial_derivative}{sub a}f{sub n}(0) = K {times} {l_brace}0{r_brace} ({prime}lim{prime} in the Kuratowski-Painleve sense). For the finite dimensional case a much stronger property holds: Given any compact subset K {improper_subset} R{sup p} there exists a lipschitzian function f : R{sup p+1} {yields} R, such that {partial_derivative}{sub a}f(0) is homeomorphic with K. This means that each topological type of a compact set may be realized by some locally lipschitzian function. For the so-called partial approximate subdifferential introduced by Journal/Thibault one even has that each compact set itself may be realized by a locally lipschitzian function. Finally it turns out that dimension of the domain of f cannot be reduced to p: For continuous f : R {yields} R the approximate subdifferential may comprise at most two connected components.
- OSTI ID:
- 36125
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
Similar Records
Generalized hessian for C{sup 1+} functions in infinite dimensional normed spaces
Sensitivity of Optimal Solutions to Control Problems for Second Order Evolution Subdifferential Inclusions