On constraint qualifications in directionally differentiable multiobjective optimization problems
RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 3, pp. 255-274.

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints such that all functions are, at least, Dini differentiable (in some cases, Hadamard differentiable and sometimes, quasiconvex). Several constraint qualifications are given in such a way that generalize both the qualifications introduced by Maeda and the classical ones, when the functions are differentiable. The relationships between them are analyzed. Finally, we give several Kuhn-Tucker type necessary conditions for a point to be Pareto minimum under the weaker constraint qualifications here proposed.

DOI : 10.1051/ro:2004023
Classification : 90C29, 90C46
Mots-clés : multiobjective optimization problems, constraint qualification, necessary conditions for Pareto minimum, Lagrange multipliers, tangent cone, Dini differentiable functions, Hadamard differentiable functions, quasiconvex functions 
@article{RO_2004__38_3_255_0,
     author = {Giorgi, Giorgio and Jim\'enez, Bienvenido and Novo, Vincente},
     title = {On constraint qualifications in directionally differentiable multiobjective optimization problems},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {255--274},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/ro:2004023},
     mrnumber = {2091756},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro:2004023/}
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Giorgi, Giorgio; Jiménez, Bienvenido; Novo, Vincente. On constraint qualifications in directionally differentiable multiobjective optimization problems. RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 3, pp. 255-274. doi : 10.1051/ro:2004023. http://www.numdam.org/articles/10.1051/ro:2004023/

[1] J.P. Aubin and H. Frankowska, Set-valued analysis. Birkhaüser, Boston (1990). | MR | Zbl

[2] M.S. Bazaraa and C.M. Shetty, Foundations of optimization. Springer-Verlag, Berlin (1976). | MR | Zbl

[3] M.S. Bazaraa and C.M. Shetty, Nonlinear programming. John Wiley & Sons, New York (1979). | MR | Zbl

[4] V.F. Demyanov and A.M. Rubinov, Constructive nonsmooth analysis. Verlag Peter Lang, Frankfurt am Main (1995). | MR | Zbl

[5] G. Giorgi and S. Komlósi, Dini derivatives in optimization. Part I. Riv. Mat. Sci. Econom. Social. Anno 15 (1992) 3-30. | Zbl

[6] Y. Ishizuka, Optimality conditions for directionally differentiable multiobjective programming problems. J. Optim. Theory Appl. 72 (1992) 91-111. | Zbl

[7] B. Jiménez and V. Novo, Cualificaciones de restricciones en problemas de optimización vectorial diferenciables. Actas XVI C.E.D.Y.A./VI C.M.A. Vol. I, Universidad de Las Palmas de Gran Canaria, Spain (1999) 727-734.

[8] B. Jiménez and V. Novo, Alternative theorems and necessary optimality conditions for directionally differentiable multiobjective programs. J. Convex Anal. 9 (2002) 97-116. | Zbl

[9] B. Jiménez and V. Novo, Optimality conditions in directionally differentiable Pareto problems with a set constraint via tangent cones. Numer. Funct. Anal. Optim. 24 (2003) 557-574. | Zbl

[10] T. Maeda, Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl. 80 (1994) 483-500. | Zbl

[11] O.L. Mangasarian, Nonlinear programming. McGraw-Hill, New York (1969). | MR | Zbl

[12] V. Novo and B. Jiménez, Lagrange multipliers in multiobjective optimization under mixed assumptions of Fréchet and directional differentiability, in 5th International Conference on Operations Research, University of La Habana, Cuba, March 4-8 (2002). Investigación Operacional 25 (2004) 34-47. | Zbl

[13] V. Preda and I. Chitescu, On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl. 100 (1999) 417-433. | Zbl

[14] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). | MR | Zbl

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