Abstract
In this paper we first provide a general formula of inclusion for the Dini-Hadamard ε-subdifferential of the difference of two functions and show that it becomes equality in case the functions are directionally approximately starshaped at a given point and a weak topological assumption is fulfilled. To this end we give a useful characterization of the Dini-Hadamard ε-subdifferential by means of sponges. The achieved results are employed in the formulation of optimality conditions via the Dini-Hadamard subdifferential for cone-constrained optimization problems having the difference of two functions as objective.
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Radu Ioan Bot’s research partially supported by DFG (German Research Foundation), project WA 922/1-3. Delia-Maria Nechita’s research done during the stay of the author in the academic year 2009/2010 at Chemnitz University of Technology as a guest of the Chair of Applied Mathematics (Approximation Theory). The author wishes to thank for the financial support provided from programs co-financed by The Sectoral Operational Programme Human Resources Development, Contract POSDRU 6/1.5/S/3—“Doctoral studies: through science towards society”.
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Boţ, R.I., Nechita, DM. On the Dini-Hadamard subdifferential of the difference of two functions. J Glob Optim 50, 485–502 (2011). https://doi.org/10.1007/s10898-010-9604-y
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DOI: https://doi.org/10.1007/s10898-010-9604-y
Keywords
- Fréchet ε-subdifferential
- Dini-Hadamard ε-subdifferential
- Sponge
- Approximately starshaped functions
- Directionally approximately starshaped functions
- Cone-constrained nonsmooth nonconvex optimization problems
- Optimality conditions in subdifferential form