Conjugate duality in set optimization with lattice structure via scalarization

From the first definitions of lower and upper type set order relations on the power set of topological vector space introduced by Kuroiwa et al. in the end of the 20th century, research on set optimization problem has developed over the last 20 years.By the definitions of equivalent classes with respect to the above set relations and certain hull operations, Hamel et al. defined spaces of sets which enjoy lattice structure.They called the above one complete lattice approach to set optimization.They also pointed out that the subset or supset inclusions appears as a partial order.In this paper, we derive weak duality theorems in the framework of set optimization problem with lattice structure, which are based on the observation that a dual optimization problem is set-valued.In order to derive strong duality theorems, we employ a nonlinear scalarizing technique for sets with lattice structure.Introducing certain family of sets, we obtain representation results in set optimization problem.The above approach allows us to derive strong duality statements.Applications to uncertain multi-objective optimization problem of the results are also given.