On approximate Pareto solutions in nonsmooth interval-valued multiobjective optimization with data uncertainty in constraints

TL;DR

This paper studies approximate Pareto solutions in nonsmooth interval-valued multiobjective optimization problems with data uncertainty, introducing new solution concepts, existence results, and duality relations under generalized convexity.

Contribution

It introduces new types of approximate Pareto solutions for the robust problem, establishes existence results using scalar penalty functions, and develops duality relations with a dual multiobjective problem.

Findings

Existence of almost regular $ ext{E}$-Pareto solutions satisfying KKT conditions.

Sufficient conditions and Wolfe-type $ ext{E}$-duality relations established.

Dual problem formulated via $ ext{E}$-interval-valued Lagrangian with duality analysis.

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.