The notions of global subdifferentials associated with the global directional derivatives are introduced in the following paper. Most common used properties, a set of calculus rules along with a mean value theorem are presented as well. In addition, a diversity of comparisons with well-known subdifferentials such as Fréchet, Dini, Clarke, Michel–Penot, and Mordukhovich subdifferential and convexificator notion are provided. Furthermore, the lower global subdifferential is in fact proved to be an abstract subdifferential. Therefore, the lower global subdifferential satisfies standard properties for subdifferential operators. Finally, two applications in nonconvex nonsmooth optimization are given: necessary and sufficient optimality conditions for a point to be local minima with and without constraints, and a revisited characterization for nonsmooth quasiconvex functions.
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This research was partially supported by Conicyt–Chile under project Fondecyt Iniciación 11180320 (Lara) and by IPM under the grant number 98900032 (Kabgani).
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Lara, F., Kabgani, A. On global subdifferentials with applications in nonsmooth optimization. J Glob Optim (2021). https://doi.org/10.1007/s10898-020-00981-1
- Nonsmooth analysis
- Global derivatives
- Global subdifferentials
- Nonconvex optimization
- Local minima
Mathematics Subject Classification