Journal ArticleDOI
Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces
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This study presents different properties of tangent cones associated with an arbitrary subset of a Banach space and establishes correlations with some of the existing results.Abstract:
This study is devoted to constrained optimization problems in Banach spaces. We present different properties of tangent cones associated with an arbitrary subset of a Banach space and establish correlations with some of the existing results. In absence of both differentiability and convexity assumptions on the functions involved in the optimization problem, the consideration of these tangent cones and their polars leads us to introduce new concepts in nondifferentiable programming. Necessary optimality conditions are first developed in a general abstract form; then these conditions are made more precise in the presence of equality constraints by introducing the concept of normal subcone.read more
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Generalized Directional Derivatives and Subgradients of Nonconvex Functions
TL;DR: The sub differential calculus (SDC) as discussed by the authors is a generalization of the distribution theory of differential equations, which associates with an extended real-valued function ǫ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ǒ at x. These form a set ∂ǫ(x) that is always convex and weak*-closed (possibly empty).
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Lipschitz Behavior of Solutions to Convex Minimization Problems
TL;DR: The Lipschitz dependence of the set of solutions of a convex minimization problem and its Lagrange multipliers upon the natural parameters from an inverse function theorem for set-valued maps is derived.
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Nonsmooth analysis: differential calculus of nondifferentiable mappings
TL;DR: In this paper, a new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested, based on the use of set-valued mappings of a special kind, called fans, for local approximation.
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Clarke's tangent cones and the boundaries of closed sets in Rn
TL;DR: Clarke as discussed by the authors defined the tangent cone as a nonempty closed subset of a non-empty subset of the Euclidean space, such that, whenever one has sequences $t_k\downarrow 0$ and $x_k \rightarrow x$ with $x-k \in C, there exist sequences with t_k + t_ky_k in C for all $k.
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Lipschitzian properties of multifunctions
TL;DR: On etudie des fonctions multiformes munies de certaines proprietes lipschitziennes On donne des relations entre les proprietes and on les exprime en termes de fonction distance.
References
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Generalized gradients and applications
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A New Approach to Lagrange Multipliers
TL;DR: The main analytical tool in this paper is an extension to infinite-dimensional spaces of the “generalized gradient” previously introduced by the author, and the calculus of the generalized gradient is explored as a preliminary step.
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On the inverse function theorem.
TL;DR: A short set of notes includes a complete proof of the Inverse Function Theorem as discussed by the authors, which is a theorem that is related to the one of the proofs in this paper.
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Generalized gradients of Lipschitz functionals
TL;DR: In this paper, the authors present a unified theory of generalized gradients, whose elements are at present scattered in various sources, and give an account of the ways in which the theory has been applied; and prove new results concerning generalized gradient of summation functionals, pointwise maxima, and integral functionals on subspaces of L∞.