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• W2016456744 abstract "IN NONSMOOTH analysis and optimization, subgradients come in many different flavors, e.g. approximate, Dini, proximal, (Clarke) generalized; see [2, 5, 6, 12, 14, 191. These subgradients are important and valuable tools. However, many questions remain unsolved concerning the exact link between the function and its subgradients. For instance, can two functions, not differing by an additive constant, have the same subgradients? In this paper, we study the fundamental problem of determining functions that can be recovered, up to an additive constant, from the knowledge of their subgradients. This “integration” problem is not very well understood, and very few functions or classes of functions are known to be recoverable from their subgradients. In Section 4, we show that if the “basic constraint qualification” holds at R, then the composition of a closed (i.e. lowersemicontinuous) proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients (actually in this case all above-mentioned subgradients are the same). Beside the obvious theoretical interest of this integration problem, it is our hope (or perhaps our long-term goal) that once this problem is better understood, we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives. An example of such an equation that well deserves study is the extended Hamilton-Jacobi equation used in optimal control; see Clarke [2]. Let us also mention that a problem similar to the integration problem is the one of determining the set-valued mappings that are in fact subgradient set-valued mappings (uniqueness is not mandatory); for a contribution to this problem see Janin [8]. Before we look at some of the known cases, where the function can be recovered from its subgradients, let us look at some negative examples. It is clear that not every function can be recovered, up to an additive constant, from its subgradients. We only need to look at the following two functions: 0 x50 0 x10 f(x) = 1 x>o g(x) = 2 x>o." @default.
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• W2016456744 date "1991-01-01" @default.
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• W2016456744 title "Integration of subdifferentials of nonconvex functions" @default.
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• W2016456744 doi "https://doi.org/10.1016/0362-546x(91)90078-f" @default.
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