Approximate solutions in set-valued optimization problems with applications to maximal monotone operators

This paper is devoted to the study of efficient elements for set-valued maps. We propose two new notions of relative weak ϵ -efficient element and strict relative weak ϵ -efficient element of set-valued maps and provide new necessary optimality conditions for the proposed concepts. We provide existence results for efficient elements. The critical ingredients for the existence results for efficient elements are the well-known separation arguments and Fan’s lemma. As an application of the existence results, we derive relationships between the efficiency concepts and the local optimizers of certain optimization problems.