The author extends the notion of strict minimum for scalar optimization problems to vector optimization problems. The notion of strict local minimum of order m and strict local minimum for vector optimization problems are introduced. Their properties and characterization are studied for multiobjective problems. Also the notion of super-strict efficiency for multiobjective problems is introduced and it is shown that these notions coincide in the scalar case. The necessary conditions for strict and super-strict minimality of order \(m\) for a multiobjective problem are stated by making use of the directional derivatives already considered by \textit{M. Studniarski} [SIAM J. Control Optim. 24, 1044--1049 (1986; Zbl 0604.49017)] and \textit{M. Studniarski} [SIAM J. Control Optim. 24, 1044--1049 (1986; Zbl 0604.49017)] and \textit{D. E. Ward} [J. Optim. Theory Appl. 80, No. 3, 551--571 (1994; Zbl 0797.90101)]. A necessary and sufficient condition for strict efficiency of order 1 for Hadamard differentiable functions is established followed by a characterization of super-strict efficiency of order 1 for Fréchet differentiable functions. The author claims that this extends to multiobjective problems th sufficient optimality conditions given in Theorem 6.3 of chapter 4 by \textit{M. R. Hestenes} [Optimization Theory: The Finite Dimensional Case, Wiley, New York (1975; Zbl 0327.90015), Krieger, Huntington (1981)].