Nguyen Thanh Qui

Nonlinear Perturbations of Polyhedral Normal Cone Mappings and Affine Variational Inequalities

This paper establishes an upper estimate for the Fréchet normal cone to the graph of the nonlinearly perturbed polyhedral normal cone mappings in finite dimensional spaces. Under a positive linear independence assumption on the normal vectors of the active constraints at the point in question, the result leads to an upper estimate for values of the Mordukhovich coderivative of such mappings. On the basis, new results on solution stability of parametric affine variational inequalities under nonlinear perturbations are derived.

Generalized Differentiation of a Class of Normal Cone Operators

This paper investigates generalized differentiation of normal cone operators to parametric smooth-boundary sets in Asplund spaces. We obtain formulas for computing the Fréchet and Mordukhovich coderivatives of such normal cone operators. We also give several examples to illustrate how the formulas can be used in practical calculations and applications.

Stability for trust-region methods via generalized differentiation

We obtain necessary and sufficient conditions for local Lipschitz-like property and sufficient conditions for local metric regularity in Robinson’s sense of Karush–Kuhn–Tucker point set maps of trust-region subproblems in trust-region methods. The main tools being used in our investigation are dual criteria for fundamental properties of implicit multifunctions which are established on the basis of generalized differentiation of normal cone mappings.

Coderivatives of implicit multifunctions and stability of variational systems

We establish formulas for computing/estimating the regular and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smooth-boundary constraint sets.

Stability of generalized equations under nonlinear perturbations

This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.

Stability for bang-bang control problems of partial differential equations

ABSTRACT In this paper, we investigate solution stability for control problems of partial differential equations with the cost functional not involving the usual quadratic term for the control. We first establish a sufficient optimality condition for the optimal control problems with bang-bang controls. Then we obtain criteria for solution stability for the optimal control problems of bang-bang controls under linear perturbations. We prove Hölder stability of optimal controls in .