N. Q. Huy

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.

Semi-Infinite Optimization under Convex Function Perturbations: Lipschitz Stability

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to illustrate the obtained results.

Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems

AbstractLet (P) denote the vector maximization problem

Exact Formulae for Coderivatives of Normal Cone Mappings to Perturbed Polyhedral Convex Sets

\max\{f(x)=\big(f_1(x),\ldots,f_m(x)\big){:}\,x\in D\},

New Second-Order Optimality Conditions for a Class of Differentiable Optimization Problems

where the objective functions fi are strictly quasiconcave and continuous on the feasible domain D, which is a closed and convex subset of Rn. We prove that if the efficient solution set E(P) of (P) is closed, disconnected, and it has finitely many (connected) components, then all the components are unbounded. A similar fact is also valid for the weakly efficient solution set Ew(P) of (P). Especially, if fi (i=1,...,m) are linear fractional functions and D is a polyhedral convex set, then each component of Ew(P) must be unbounded whenever Ew(P) is disconnected. From the results and a result of Choo and Atkins [J. Optim. Theory Appl. 36, 203–220 (1982.)] it follows that the number of components in the efficient solution set of a bicriteria linear fractional vector optimization problem cannot exceed the number of unbounded pseudo-faces of D.

Stability of Implicit Multifunctions in Banach Spaces

This paper is devoted to developing new applications from the limiting subdifferential in nonsmooth optimization and variational analysis to the study of the Lipschitz behavior of the Pareto solution maps in parametric nonconvex semi-infinite vector optimization problems (SIVO for brevity). We establish sufficient conditions for the Aubin Lipschitz-like property of the Pareto solution maps of SIVO under perturbations of both the objective function and constraints.

Strong second-order Karush–Kuhn–Tucker optimality conditions for vector optimization

In this paper, without using any regularity assumptions, we derive a new exact formula for computing the Fréchet coderivative and an exact formula for the Mordukhovich coderivative of normal cone mappings to perturbed polyhedral convex sets. Our development establishes generalizations and complements of the existing results on the topic. An example to illustrate formulae is given.

CODERIVATIVES OF FRONTIER AND SOLUTION MAPS IN PARAMETRIC MULTIOBJECTIVE OPTIMIZATION

In the present paper, we establish some results for the existence of optimal solutions in vector optimization in infinite-dimensional spaces, where the optimality notion is understood in the sense of generalized order (may not be convex and/or conical). This notion is induced by the concept of set extremality and covers many of the conventional notions of optimality in vector optimization. Some sufficient optimality conditions for optimal solutions of a class of vector optimization problems, which satisfies the free disposal hypothesis, are also examined.