Abstract
Construction of a united theory of subdifferentials of the first and second orders is interesting for many specialists in optimization. In this paper, the rules for construction of subdifferentials of the first and second orders are introduced. The constructions are done with the help of the Steklov integrals of Lipschitz functions over the images of set-valued mappings. It is proved that the subdifferential of the first order, consisting of the average integral limit values of the gradients of a Lipschitz function, calculated along the curves from an introduced set of the curves, coincides with the subdifferential of the first order, introduced by the author, constructed using the Steklov integral. If a function is twice differentiable at some point, then subdifferentials of the first and second orders coincide with the gradient and the matrix of the second mixed derivatives of this function at the same point. The generalized gradients and matrices are used for formulation of the necessary and sufficient conditions of optimality. The calculus for subdifferentials of the first and second orders is constructed. The examples are given.
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Dedicated to the memory of my teacher Prof. V. F. Demyanov, who formulated the problem about the subdifferential of the second order long time ago.
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Prudnikov, I.M. Subdifferentials of the First and Second Orders for Lipschitz Functions. J Optim Theory Appl 171, 906–930 (2016). https://doi.org/10.1007/s10957-016-1010-2
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DOI: https://doi.org/10.1007/s10957-016-1010-2
Keywords
- Lipschitz functions
- Set-valued Mappings
- Generalized gradients and matrices
- Necessary and sufficient conditions of optimality
- Steklov integral