Approximate solutions in multiobjective interval-valued optimization problems: Existence theorems and optimality conditions
Chuang-liang Zhang, Yun-cheng Liu, Nan-jing Huang
TL;DR
This paper addresses approximate solutions for a multiobjective interval-valued optimization problem (MIOP) formulated with the interval order $\prec_{CW}$. It introduces an interval-valued Ekeland variational principle and proves existence results for weak $\varepsilon$-minimal and weak $\varepsilon$-quasi-minimal solutions, even in nonsmooth and nonconvex settings. Utilizing nonsmooth analysis, it derives both necessary and sufficient KKT-type optimality conditions, including approximate and modified $\epsilon$-KKT points, and extends the framework to multiplayer games for approximate Nash equilibria. The results provide a theoretical foundation for optimization under interval uncertainty and offer a pathway to applying these techniques in interval-valued game theory.
Abstract
This paper is devoted to the study of approximate solutions for a multiobjective interval-valued optimization problem based on an interval order. We establish new existence theorems of approximate solutions for such a problem under some mild conditions. Moreover, we give KKT optimality conditions for approximate solutions for such a problem whose associated functions are nonsmooth and nonconvex. We also propose the approximate KKT optimality condition of an approximate solution for such a problem. Finally, we apply some obtained results to a noncooperative game involving the multiobjective interval-valued function.