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 differentiat...

We employ a fuzzy optimality condition for the Fréchet subdifferential and some advanced techniques of variational analysis such as formulae subdifferentials an infinite family nonsmooth functions coderivative scalarization to investigate robust duality nonsmooth/nonconvex multiobjective optimization problem dealing with uncertain constraints in arbitrary Asplund spaces. establish necessary con...

We investigate sample average approximation of a general class of onestage stochastic mathematical programs with equilibrium constraints. By using graphical convergence of unbounded set-valued mappings, we demonstrate almost sure convergence of a sequence of stationary points of sample average approximation problems to their true counterparts as the sample size increases. In particular we show ...

A new framework for studying superspace is given, based on methods from Clifford analysis. This leads to the introduction of both orthogonal and symplectic Clifford algebra generators, allowing for an easy and canonical introduction of a super-Dirac operator, a super-Laplace operator and the like. This framework is then used to define a super-Hodge coderivative, which, together with the exterio...

In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number canbe estimated fromabove explicitly in termsof the proble...

A condition ensuring metric subregularity (respectively calmness) of general multifunctions between Banach spaces is derived. In finite dimensions this condition can be expressed in terms of a derivative which appears to be a combination of the coderivative and the contingent derivative. It is further shown that this sufficient conditions is in some sense the weakest possible first-order condit...

In this paper we perform sensitivity analysis for optimization problems with variational inequality constraints (OPVICs). We provide upper estimates for the limiting subdifferential (singular limiting subdifferential) of the value function in terms of the set of normal (abnormal) coderivative (CD) multipliers for OPVICs. For the case of optimization problems with complementarity constraints (OP...