On the existence and the stability of solutions in nonconvex vector optimization
Tran Van Nghi, Le Ngoc Kien, Nguyen Van Tuyen
TL;DR
The paper studies existence and stability of weak Pareto solutions for constrained nonconvex vector optimization with unbounded feasibility via asymptotic tools. By employing asymptotic cones $X^{\infty}$ and generalized $q$-asymptotic functions $f^{\infty}_{q}$, it derives conditions ensuring nonempty, compact weak Pareto solutions and a weak sharp minima at infinity property, together with coercivity of scalarized objectives. It further examines stability under linear perturbations, proving upper/lower semicontinuity and closedness of the weak Pareto solution map, and extends results to the class of $\mathbb{R}^m_+$-robustly quasiconvex vector objectives using $q$-asymptotic analysis. The framework yields practical criteria for solution existence, stability, and robustness in nonconvex multiobjective problems, with applications to robust quasiconvex vector optimization.
Abstract
The paper is devoted to the existence of weak Pareto solutions and the weak sharp minima at infinity property for a general class of constrained nonconvex vector optimization problems with unbounded constraint set via asymptotic cones and generalized asymptotic functions. Then we show that these conditions are useful for studying the solution stability of nonconvex vector optimization problems with linear perturbation. We also provide some applications for a subclass of robustly quasiconvex vector optimization problems.