The aim is to investigate the optimality conditions for a specific class of multi-objective optimization problems with a norm structure. The Fritz-John and Kuhn-Tucker necessary optimality conditions for the Pareto optimum are systematically examined by calculating the subdifferential of the objective functions of these problems. Theorems are utilized to establish the definitions of novel Pareto-FJ and Pareto-KKT stationary points. Additionally, equivalence conditions for these stationary points, together with their related geometric optimality conditions, are presented and then proven. Several theorems such as the Pareto-FJ optimality condition and the Pareto-KKT optimality condition are established for a class of norm-structured convex multi-objective optimization problems with general constraints and interval constraints. The results obtained enrich the theory of multi-objective optimization and provide a foundational basis for the applied research of multi-objective optimization problems with a normative structure.