Huynh Van Ngai

Metric Inequality, Subdifferential Calculus and Applications

In this paper we establish characterizations of Asplund spaces in terms of conditions ensuring the metric inequality and intersection formulae. Then we establish chain rules for the limiting Fréchet subdifferentials. Necessary conditions for constrained optimization problems with non-Lipschitz data are derived.

Error Bounds and Implicit Multifunction Theorem in Smooth Banach Spaces and Applications to Optimization

In this paper we provide an error bound estimate and an implicit multifunction theorem in terms of smooth subdifferentials and abstract subdifferentials. Then, we derive a subdifferential calculus and Fritz–John type necessary optimality conditions for constrained minimization problems.

Error bounds for systems of lower semicontinuous functions in Asplund spaces

In this paper, using the Fréchet subdifferential, we derive several sufficient conditions ensuring an error bound for inequality systems in Asplund spaces. As an application we obtain in the context of Banach spaces a global error bound for quadratic nonconvex inequalities and we derive necessary optimality conditions for optimization problems.

Error Bounds in Metric Spaces and Application to the Perturbation Stability of Metric Regularity

This paper was motivated by the need to establish some new characterizations of the metric regularity of set-valued mappings. Through these new characterizations it was possible to investigate the global/local perturbation stability of the metric regularity and to extend a result by Ioffe [Set-Valued Anal., 9 (2001), pp. 101-109] on the perturbation stability of the global metric regularity when the image space is not necessarily complete. It was also possible to give a characterization of the local metric regularity and to derive a local version of the perturbation stability of the metric regularity. In this work we also describe an application of this perturbation stability and give a simple proof of a result on the error bound of 2-regular mappings established by Izmailov and Solodov [Math. Program., 89 (2001), pp. 413-435] and generalized by He and Sun [Math. Oper. Res., 30 (2005), pp. 701-717].

Stability of Error Bounds for Convex Constraint Systems in Banach Spaces

This paper studies stability of error bounds for convex constraint systems in Banach spaces. We show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one. A single inequality as well as semi-infinite constraint systems are considered.

Stability of Error Bounds for Semi-infinite Convex Constraint Systems

In this paper, we are concerned with the stability of the error bounds for semi-infinite convex constraint systems. Roughly speaking, the error bound of a system of inequalities is said to be stable if all its “small” perturbations admit a (local or global) error bound. We first establish subdifferential characterizations of the stability of error bounds for semi-infinite systems of convex inequalities. By applying these characterizations, we extend some results established by Aze and Corvellec [SIAM J. Optim., 12 (2002), pp. 913-927] on the sensitivity analysis of Hoffman constants to semi-infinite linear constraint systems.

Error bounds for convex differentiable inequality systems in Banach spaces

The paper is devoted to studying the Hoffman global error bound for convex quadratic/affine inequality/equality systems in the context of Banach spaces. We prove that the global error bound holds if the Hoffman local error bound is satisfied for each subsystem at some point of the solution set of the system under consideration. This result is applied to establishing the equivalence between the Hoffman error bound and the Abadie qualification condition, as well as a general version of Wang & Pangs result [30], on error bound of Hölderian type. The results in the present paper generalize and unify recent works by Luo & Luo in [17], Li in [16] and Wang & Pang in [30].

Metric Regularity of the Sum of Multifunctions and Applications

The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported.

Newton's Method for Solving Inclusions Using Set-Valued Approximations

Results on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed.

Directional metric regularity of multifunctions

In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.