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Zbl 1010.90075
Jiménez, Bienvenido
Strict efficiency in vector optimization.
(English)
[J] J. Math. Anal. Appl. 265, No.2, 264-284 (2002). ISSN 0022-247X

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 {\it M. Studniarski} [SIAM J. Control Optim. 24, 1044--1049 (1986; Zbl 0604.49017)] and {\it M. Studniarski} [SIAM J. Control Optim. 24, 1044--1049 (1986; Zbl 0604.49017)] and {\it D. E. Ward} [J. Optim. Theory Appl. 80, No. 3, 551--571 (1994; Zbl 0797.90101)].\par 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 {\it M. R. Hestenes} [Optimization Theory: The Finite Dimensional Case, Wiley, New York (1975; Zbl 0327.90015), Krieger, Huntington (1981)].
[R.N.Kaul (Delhi)]
MSC 2000:
*90C29 Multi-objective programming, etc.
90C46 Optimality conditions, duality

Keywords: vector optimization; strict minimum; super-strict minimum; optimality conditions

Citations: Zbl 0797.90101; Zbl 0327.90015; Zbl 0604.49017

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