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 for a multiobjective problem are stated by making use of the directional derivatives already considered by M. Studniarski [SIAM J. Control Optim. 24, 1044–1049 (1986; Zbl 0604.49017)] and M. Studniarski [SIAM J. Control Optim. 24, 1044–1049 (1986; Zbl 0604.49017)] and 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 M. R. Hestenes [Optimization Theory: The Finite Dimensional Case, Wiley, New York (1975; Zbl 0327.90015), Krieger, Huntington (1981)].