Nguyen Dong Yen

Vector variational inequality as a tool for studying vector optimization problems

The paper aims to show that a Vector Variational Inequality can be an useful tool for studying a Vector Optimization Problem.

Holder continuity of solutions to a parametric variational inequality

We prove a Hölder continuity property of the locally unique solution to a parametric variational inequality without assuming differentiability of the given data.

Subgradients of marginal functions in parametric mathematical programming

In this paper we derive new results for computing and estimating the so-called Fréchet and limiting (basic and singular) subgradients of marginal functions in real Banach spaces and specify these results for important classes of problems in parametric optimization with smooth and nonsmooth data. Then we employ them to establish new calculus rules of generalized differentiation as well as efficient conditions for Lipschitzian stability and optimality in nonlinear and nondifferentiable programming and for mathematical programs with equilibrium constraints. We compare the results derived via our dual-space approach with some known estimates and optimality conditions obtained mostly via primal-space developments.

Lipschitz Continuity of Solutions of Variational Inequalities with a Parametric Polyhedral Constraint

It is proved that the metric projection from a point onto a moving polyhedron is Lipschitz continuous with respect to the perturbations on the right-hand sides of the linear inequalities defining the polyhedron. The property leads to a simple sufficient condition for Lipschitz continuity of a locally unique solution of parametric variational inequalities with a moving polyhedral constraint set. Applications of these results to traffic network equilibrium problems are given in detail.

Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming

We develop various (exact) calculus rules for Fréchet lower and upper subgradients of extended-real-valued functions in real Banach spaces. Then we apply this calculus to derive new necessary optimality conditions for some remarkable classes of problems in constrained optimization including minimization problems for difference-type functions under geometric and operator constraints as well as subdifferential optimality conditions for the so-called weak sharp minima. §Dedicated to Diethard Pallaschke in honor of his 65th birthday.

Correction: On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints

In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others.

Solution Stability of Nonsmooth Continuous Systems with Applications to Cone-Constrained Optimization

In this paper we establish conditions for stability, metric regularity, and a pseudo-Lipschitz property of the solution maps of parametric inequality systems involving nonsmooth (not necessarily locally Lipschitz) continuous functions and closed convex sets. We also derive open mapping and inverse mapping theorems for nonsmooth continuous functions, Lagrange multiplier rules for nonsmooth cone-constrained optimization problems, and conditions for the continuity of the optimal value functions of optimization problems. The main tool used is a generalized Jacobian, called approximate Jacobian. It provides a flexible nonsmooth local analysis of continuous functions and often gives sharp calculus rules for locally Lipschitz functions. The regularity condition, which plays a key role in the local analysis, is a new extension of the Robinson regularity condition for continuous functions.

Stability of the solution set of perturbed nonsmooth inequality systems and application

Stability properties of the solution set of generalized inequality systems with locally Lipschitz functions are obtained under a regularity condition on the generalized Jacobian and the Clarke tangent cone. From these results, we derive sufficient conditions for the optimal value function in a nonsmooth optimization problem to be continuous or locally Lipschitz at a given parameter.

On the solution existence of pseudomonotone variational inequalities

As shown by Thanh Hao [Acta Math. Vietnam 31, 283–289, 2006], the solution existence results established by Facchinei and Pang [Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I (Springer, Berlin, 2003) Prop. 2.2.3 and Theorem 2.3.4] for variational inequalities (VIs) in general and for pseudomonotone VIs in particular, are very useful for studying the range of applicability of the Tikhonov regularization method. This paper proposes some extensions of these results of Facchinei and Pang to the case of generalized variational inequalities (GVI) and of variational inequalities in infinite-dimensional reflexive Banach spaces. Various examples are given to analyze in detail the obtained results.

On the stability of solutions to quadratic programming problems

Abstract.We consider the parametric programming problem (Qp) of minimizing the quadratic function f(x,p):=xTAx+bTx subject to the constraint Cx≤d, where x∈ℝn, A∈ℝn×n, b∈ℝn, C∈ℝm×n, d∈ℝm, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Qp) are denoted by M(p) and Mloc(p), respectively. It is proved that if the point-to-set mapping Mloc(·) is lower semicontinuous at p then Mloc(p) is a nonempty set which consists of at most ?m,n points, where ?m,n=