Generalized weak ε-subdifferential and applications

A concept of subdifferential of a vector-valued mapping is introduced, called generalized weak ε-subdifferential.We establish existence theorems and investigate their main properties, and provide illustrative examples to clarify the construction. This construction extends and unifies several existing notions of approximate subgradients in vector optimization, including the Pareto weak subdifferential. We establish some formulas of the generalized weak ε-subdifferential for the sum and the difference of two vector-valued mappings. A relationship between the generalized weak ε-subdifferential and a directional derivative is presented. We discuss the positive homogeneity of the generalized weak ε-subdifferential. As application of the calculus rules, we establish necessary and sufficient optimality conditions for a constrained vector optimization problem with the difference of two vector-valued mappings.