Gradients on sets. (English) Zbl 1452.46034
Summary: For a locally Lipschitz continuous function \(f\colon X\to\mathbb{R}\) the generalized gradient \(\partial f(x)\) of Clarke is used to develop some (set-valued) gradient on a set \(A\subset X\). Existence, uniqueness and some approximation are considered for optimal descent directions on set \(A\). The results serve as basis for nonsmooth numerical descent algorithms that can be found in subsequent papers.
MSC:
| 46G05 | Derivatives of functions in infinite-dimensional spaces |
| 49J52 | Nonsmooth analysis |
| 46T20 | Continuous and differentiable maps in nonlinear functional analysis |
| 47H04 | Set-valued operators |
| 49K99 | Optimality conditions |
| 65K10 | Numerical optimization and variational techniques |
Keywords:
generalized gradient; set-valued gradient; generalized directional derivative; Lipschitz continuous function; optimal descent directionReferences:
| [1] | J. Aubin, I. Ekeland:Applied Nonlinear Analysis, Wiley, New York (1984). · Zbl 0641.47066 |
| [2] | I. Cioranescu:Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht (1990). · Zbl 0712.47043 |
| [3] | F. H. Clarke:Optimization and Nonsmooth Analysis, Wiley, New York (1983). 1070J. Mankau, F. Schuricht / Gradients on Sets · Zbl 0582.49001 |
| [4] | F. H. Clarke:Functional Analysis, Calculus of Variations and Optimal Control, Springer, London (2013). · Zbl 1277.49001 |
| [5] | A. A. Goldstein:Optimization of Lipschitz continuous functions, Math. Programming 13 (1977) 14-22. · Zbl 0394.90088 |
| [6] | D. Werner:Funktionalanalysis, |
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