A Derivative-coderivative Inclusion in Second-order Nonsmooth Analysis
نویسنده
چکیده
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.
منابع مشابه
Study of a Coeecient Control Problem Using Nonsmooth Analysis
This report is concerned with a control problem associated to a semilinear second order ordinary diierential equation with pointwise state constraints. The control acts as a coeecient of the state equation. The nonlinear part of the equation is governed by a Nemytskij operator deened by a nonsmooth function. We prove the existence of optimal controls and obtain a necessary optimality condition ...
متن کاملNonsmooth Lyapunov Pairs for Infinite-dimensional First-order Differential Inclusions∗
The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by the means of the proximal subdifferential of the no...
متن کاملA necessary second-order optimality condition in nonsmooth mathematical programming
Generalized second–order directional derivatives for nonsmooth real–valued functions are studied and their connections with second–order variational sets are investigated. A necessary second–order optimality condition for problems with inequality constraints is obtained.
متن کاملAnalysis of nonsmooth vector-valued functions associated with second-order cones
Let Kn be the Lorentz/second-order cone in R. For any function f from R to R, one can define a corresponding function f soc(x) on R by applying f to the spectral values of the spectral decomposition of x ∈ R with respect to Kn. We show that this vector-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiabi...
متن کاملOn second – order conditions in vector optimization
Starting from second-order conditions for C scalar unconstrained optimization problems described in terms of the second-order Dini directional derivative, we pose the problem, whether similar conditions for C vector optimization problems can be derived. We define second-order Dini directional derivatives for vector functions and apply them to formulate such conditions as a Conjecture. The proof...
متن کامل