On set less relations in near vector space

This paper addresses a new class of set optimization problems (SOPs) where the objective is a set-valued map defined on a vector space but taking values in a real near vector space. Unlike vector spaces, real near vector spaces lack additive inverses, which present unique challenges. To handle this, we introduce five novel set less relations on P(X), the power set of a real near vector space X, and define corresponding notions of minimizers. We also establish two types of semicontinuity for set-valued maps into real near vector spaces and prove existence results of solutions of SOP. Finally, by introducing a scalarization approach, we derive sufficient optimality conditions for the solutions of SOP in this new setting.