On approximate Pareto solutions in nonsmooth interval-valued multiobjective optimization with data uncertainty in constraints
Vu Hong Quan, Duong Thi Viet An, Nguyen Van Tuyen
TL;DR
This paper addresses robust, nonsmooth, interval-valued multiobjective optimization with data uncertainty in constraints by introducing $\mathcal{E}$-approximate Pareto notions under the $LU$ interval order. It combines Ekeland's variational principle and subdifferential calculus to derive KKT-type necessary conditions for almost $\mathcal{E}$-Pareto and almost $\mathcal{E}$-quasi Pareto solutions, and constructs a penalized scalar problem to ensure existence. Under generalized convexity, it develops Wolfe-type $\mathcal{E}$-duality and a dual multiobjective problem based on an $\mathcal{E}$-interval-valued Lagrangian, establishing both $\mathcal{E}$-duality and converse-like duality results. The work further introduces quasi-$\mathcal{E}$ Pareto saddle points and demonstrates their relation to primal solutions, providing a cohesive framework for analysis of robust interval-valued MOPs with uncertainty in the constraints.
Abstract
This paper deals with approximate Pareto solutions of a nonsmooth interval-valued multiobjective optimization problem with data uncertainty in constraints. We first introduce some kinds of approximate Pareto solutions for the robust counterpart (RMP) of the problem in question by considering the lower-upper interval order relation including: (almost, almost regular) $\mathcal{E}$-Pareto solution and (almost, almost regular) $\mathcal{E}$-quasi Pareto solution. By using a scalar penalty function, we obtain a result on the existence of an almost regular $\mathcal{E}$-Pareto solution of (RMP) that satisfies the Karush--Kuhn--Tucker necessary optimality condition up to a given precision. We then establish sufficient conditions and Wolfe-type $\mathcal{E}$-duality relations for approximate Pareto solutions of (RMP) under the assumption of generalized convexity. In addition, we present a dual multiobjective problem to the primal one via the $\mathcal{E}$-interval-valued vector Lagrangian function and examine duality relations.