On global subdifferentials with applications in nonsmooth optimization |
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Authors: | Lara Felipe Kabgani Alireza |
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Affiliation: | 1.Departamento de Matemática, Facultad de Ciencias, Universidad de Tarapacá, Arica, Chile ;2.Mathematics Group, Department of Environment, Urmia University of Technology, Urmia, Iran ;3.School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran ; |
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Abstract: | 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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