Nguyen Van Tuyen

Existence Theorems in Vector Optimization with Generalized Order

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.

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

In the present paper, we focus on the optimization problems, where objective functions are Fréchet differentiable, and whose gradient mapping is locally Lipschitz on an open set. We introduce the concept of second-order symmetric subdifferential and its calculus rules. By using the second-order symmetric subdifferential, the second-order tangent set and the asymptotic second-order tangent cone, we establish some second-order necessary and sufficient optimality conditions for optimization problems with geometric constraints. Examples are given to illustrate the obtained results.

A note on second-order Karush–Kuhn–Tucker necessary optimality conditions for smooth vector optimization problems

The aim of this note is to present some second-order Karush–Kuhn–Tucker necessary optimality conditions for vector optimization problems, which modify the incorrect result in ((), Thm. 3.2).

On the existence of Pareto solutions for polynomial vector optimization problems

We are interested in the existence of Pareto solutions to the vector optimization problem

New Second-Order Karush–Kuhn–Tucker Optimality Conditions for Vector Optimization

\begin{aligned} \mathrm{Min}_{\,{\mathbb {R}}^m_+} \{f(x) \,|\, x\in {\mathbb {R}}^n\}, \end{aligned}

Locally Lipschitz Vector Optimization Problems: Second-order Constraint Qualifications, Regularity Condition and KKT Necessary Optimality Conditions.

MinR+m{f(x)|x∈Rn},where

C^1

f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^m