We show that in two dimensions or higher, the Mordukhovich-Ioffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Fréchet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be nonconvex almost everywhere for Fréchet differentiable vector-valued Lipschitz functions. Finally we show that for vector-valued Lipschitz functions the approximate Jacobian associated with the Mordukhovich-Ioffe coderivative can be almost everywhere disconnected.
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Research supported by NSERC and the Shrum Endowment at Simon Fraser University.
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Borwein, D., Borwein, J.M. & Wang, X. Approximate subgradients and coderivatives in R n . Set-Valued Anal 4, 375–398 (1996). https://doi.org/10.1007/BF00436112
Mathematics Subject Classifications (1991)
- Primary 49J52
- Secondary 26A27
- generalized Jacobian
- Lipschitz function
- bump function
- nowhere dense set
- Lebesgue measure