Constraint specifications and their strong KKT conditions are important topics in the study of multi-objective optimization problems, and a series of important results have been obtained in this area at home and abroad. This paper focuses on nonsmooth multi-objective optimization problems containing inequality constraints and arbitrary set constraints, with both objective and constraint functions being locally Liepschitz. A new generalized Abadie constraint specification is proposed in the paper, which leads to some new results on the necessary conditions for the strong KKT optimality of the effective solution. The proof of the theorem mainly uses the strong separation theorem for convex sets.