Differentials and Semidifferentials for Metric Spaces of Shapes and Geometries

Abstract : The Hadamard semidifferential retains the advantages of the differential calculus such as the chain rule and semiconvex functions are Hadamard semidifferentiable. The semidifferential calculus extends to subsets of $${\mathbb {R}}^n$$ without Euclidean smooth structure. This set-up is an ideal tool to study the semidifferentiability of objective functions with respect to families of sets which are non-linear non-convex complete metric spaces. Shape derivatives are differentials for spaces endowed with Courant metrics. Topological derivatives are shown to be semidifferentials on the group of Lebesgue measurable characteristic functions.
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Michel Delfour. Differentials and Semidifferentials for Metric Spaces of Shapes and Geometries. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. pp.230-239, ⟨10.1007/978-3-319-55795-3_21⟩. ⟨hal-01626899⟩

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