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Strict efficiency in vector optimization. (English) Zbl 1010.90075

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 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)].

Reviewer: R.N.Kaul (Delhi)

##### MSC:
 90C29 Multi-objective programming; goal programming 90C46 Optimality conditions, duality