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)


90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming
Full Text: DOI


[1] Aubin, J. P.; Frankowska, H., Set-valued analysis (1990), Birkhaüser: Birkhaüser Boston · Zbl 0713.49021
[2] Auslender, A., Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim., 22, 239-254 (1984) · Zbl 0538.49020
[3] Ben-Tal, A.; Zowe, J., A unified theory of first and second order conditions for extremum problems in topological vector spaces, Math. Progr. Stud., 19, 39-76 (1982) · Zbl 0494.49020
[4] Cambini, R., Second order optimality conditions in multiobjective programming, Optimization, 44, 139-160 (1998) · Zbl 0917.90264
[5] Cromme, L., Strong uniqueness: A far reaching criterion for the convergence of iterative procedures, Numer. Math., 29, 179-193 (1978) · Zbl 0352.65012
[6] Gähler, S., A generalization of an optimality theorem, Nonlinear Analysis. Nonlinear Analysis, Col. Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech (1981), Akademie Verlag: Akademie Verlag Berlin, p. 347-349 · Zbl 0506.90077
[7] Giorgi, G.; Guerraggio, A., On the notion of tangent cone in mathematical programming, Optimization, 25, 11-23 (1992) · Zbl 0817.90108
[8] Hestenes, M. R., Optimization Theory: The Finite Dimensional Case (1981), Krieger: Krieger New York
[9] Sawaragi, Y.; Nakayama, H.; Tanino, T., Theory of Multiobjective Optimization (1985), Academic Press: Academic Press Orlando · Zbl 0566.90053
[10] Smale, S., Global analysis and economics V, Pareto theory with constraints, J. Math. Econom., 1, 213-221 (1974) · Zbl 0357.90010
[11] Smale, S., Sufficient Conditions for an Optimum. Sufficient Conditions for an Optimum, Lecture Notes in Math., 468 (1975), Springer-Verlag: Springer-Verlag Berlin, p. 287-292 · Zbl 0314.58007
[12] Studniarski, M., Necessary and sufficient conditions for isolated local minima of non-smooth functions, SIAM J. Control Optim., 24, 1044-1049 (1986) · Zbl 0604.49017
[13] Ursescu, C., Tangents sets calculus and necessary conditions for extremality, SIAM J. Control Optim., 20, 563-574 (1982) · Zbl 0488.49009
[14] Van Geldrop, J. H., A note on local Pareto optima, J. Math. Econom., 7, 51-54 (1980) · Zbl 0428.90005
[15] Wan, Y. H., On local Pareto optima, J. Math. Econom., 2, 35-42 (1975) · Zbl 0309.90049
[16] Ward, D. E., Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl., 80, 551-571 (1994) · Zbl 0797.90101
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