Fuzzy calculus for coderivatives of multifunctions is studied. The generalized differentiation of set-valued mappings between Banach spaces is explored. Appropriative derivative-like concepts are derived to provide an...
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Fuzzy calculus for coderivatives of multifunctions is studied. The generalized differentiation of set-valued mappings between Banach spaces is explored. Appropriative derivative-like concepts are derived to provide an effective study of local behavior of multifunctions with successive applications. One of the advantages of the coderivative is a rich calculus supported by effective characterizations of the Lipschitzian. Comprehensive results for Frechet coderivatives are obtained.
The behavior of a minimizing point when an objective function is tilted by adding a small linear term is studied from the perspective of second-order conditions for local optimality. The classical condition of a posit...
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The behavior of a minimizing point when an objective function is tilted by adding a small linear term is studied from the perspective of second-order conditions for local optimality. The classical condition of a positive-definite Hessian in smooth problems without constraints is found to have an exact counterpart much more broadly in the positivity of a certain generalized Hessian mapping. This fully characterizes the case where tilt perturbations cause the minimizing point to shift in a Lipschitzian manner.
For twice smooth functions, the symmetry of the matrix of second partial derivatives is automatic and can be seen as the symmetry of the Jacobian matrix of the gradient mapping. For nonsmooth functions, possibly even ...
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For twice smooth functions, the symmetry of the matrix of second partial derivatives is automatic and can be seen as the symmetry of the Jacobian matrix of the gradient mapping. For nonsmooth functions, possibly even extended-real-valued, the gradient mapping can be replaced by a subgradient mapping, and generalized second derivative objects can then be introduced through graphical differentiation of this mapping, but the question of what analog of symmetry might persist has remained open. An answer is provided here in terms of a derivative-coderivative inclusion.
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