Thai Doan Chuong

Pseudo-Lipschitz property of linear semi-infinite vector optimization problems

This paper is devoted to the study of the pseudo-Lipschitz property of Pareto solution map for the parametric linear semi-infinite vector optimization problem (LSVO). We establish new sufficient conditions for the pseudo-Lipschitz property of the Pareto solution map of (LSVO) under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. Examples are given to illustrate the results obtained.

Subdifferentials of Marginal Functions in Semi-infinite Programming

This paper proposes two new constraint qualification conditions (CQs) which are useful for a unified study of CQs from both a convex analysis and a nonsmooth analysis point of view. Our CQs cover the existing CQs of Mangasarian-Fromovitz and Farkas-Minkowski types. Some sufficient conditions for the validity of the new CQs are given. Under these CQs, we derive formulae for computing and/or estimating the (basic and singular) subdifferentials of marginal/optimal value function in semi-infinite programming from some results of modern variational analysis and generalized differentiation. Examples are given to illustrate the obtained formulae.

Nonsmooth Semi-infinite Multiobjective Optimization Problems

We apply some advanced tools of variational analysis and generalized differentiation to establish necessary conditions for (weakly) efficient solutions of a nonsmooth semi-infinite multiobjective optimization problem (SIMOP for brevity). Sufficient conditions for (weakly) efficient solutions of a SIMOP are also provided by means of introducing the concepts of (strictly) generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. In addition, we propose types of Wolfe and Mond–Weir dual problems for SIMOPs, and explore weak and strong duality relations under assumptions of (strictly) generalized convexity. Examples are also designed to analyze and illustrate the obtained results.

Calmness of efficient solution maps in parametric vector optimization

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.

Isolated and Proper Efficiencies in Semi-Infinite Vector Optimization Problems

This paper deals with a nonsmooth semi-infinite multiobjective/vector optimization problem (SIMOP, for short). We first establish necessary and sufficient conditions for (local) strongly isolated solutions and (local) positively properly efficient solutions of an SIMOP. Then, we propose a dual problem to the SIMOP under consideration and examine weak and strong duality relations between them.

Optimality conditions and duality in nonsmooth multiobjective optimization problems

Exploiting some tools of modern variational analysis involving the approximate extremal principle, the fuzzy sum rule for the Fréchet subdifferential, the sum rule for the limiting subdifferential and the scalarization formulae of the coderivatives, we establish necessary conditions for (weakly) efficient solutions of a multiobjective optimization problem with inequality and equality constraints. Sufficient conditions for (weakly) efficient solutions of an aforesaid problem are also provided by means of employing L-(strictly) invex-infine functions defined in terms of the limiting subdifferential. In addition, we introduce types of Wolfe and Mond–Weir dual problems and investigate weak/strong duality relations.

Steepest descent methods for critical points in vector optimization problems

In this article, we present steepest descent methods for finding stationary (critical) points of vector optimization problems for maps from an Euclidean space to a Banach space with respect to the partial order induced by a closed, convex and pointed cone with a nonempty interior. Convergence of the generated sequence to a weakly efficient solution of our problem is established under some reasonable additional hypotheses.

Hölder-Like Property and Metric Regularity of a Positive-Order for Implicit Multifunctions

We introduce concepts of metric regularity and metric subregularity of a positive-order for an implicit multifunction and provide new sufficient conditions for the implicit multifunctions to achieve the addressed properties. The conditions provided are presented in terms of the Frechet/Mordukhovich coderivative of the corresponding parametric multifunction formulated the implicit multifunction. We show that such sufficient conditions are also necessary for the metric regularity/subregularity of a positive-order of the implicit multifunction when the corresponding parametric multifunction is (locally) convex and closed. In this way, we establish criteria ensuring that an implicit multifunction is Holder-like and calm of a positive-order at a given point. As applications, we derive sufficient conditions in terms of coderivatives for a multifunction (resp., its inverse multifunction) to have the open covering property and the metric regularity/subregularity of a positive-order (resp., the Holder-like/calm property).

Normal subdifferentials of efficient point multifunctions in parametric vector optimization

The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531–562, 2007), by using the Mordukhovich coderivative of the associated epigraphical multifunction, which has proven to be useful in deriving necessary conditions for super efficient points of vector optimization problems. In this paper, we establish new formulae for computing and/or estimating the normal subdifferential of the efficient point multifunctions of parametric vector optimization problems. These formulae will be presented in a broad class of conventional vgector optimization problems with the presence of geometric, operator, equilibrium, and (finite and infinite) functional constraints.

Directionally generalized differentiation for multifunctions and applications to set-valued programming problems

The aim of this work is twofold. First, we establish sum rules for the directionally coderivatives of multifunctions and intersection rules for the directionally limiting normal cones. Then, we apply the provided formulas to derive directionally necessary conditions for a set-valued optimization problem with equilibrium constraints.