Approximate subgradients and coderivatives in Rn |
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Authors: | D. Borwein J. M. Borwein Xianfu Wang |
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Affiliation: | (1) Department of Mathematics, University of Western Ontario, N6A 5B7 London, Ontario, Canada;(2) Department of Mathematics and Statistics, Simon Fraser University, V5A 1S6 Burnaby, BC, Canada;(3) Department of Mathematics and Statistics, Simon Fraser University, V5A 1S6 Burnaby, BC, Canada |
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Abstract: | 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.Research supported by NSERC and the Shrum Endowment at Simon Fraser University. |
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Keywords: | Primary 49J52 Secondary 26A27 26B12 49J50 52A20 |
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