On constraint qualifications in directionally differentiable multiobjective optimization problems

Giorgio Giorgi; Bienvenido Jiménez; Vincente Novo

RAIRO - Operations Research (2010)

  • Volume: 38, Issue: 3, page 255-274
  • ISSN: 0399-0559

Abstract

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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.

How to cite

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Giorgi, Giorgio, Jiménez, Bienvenido, and Novo, Vincente. "On constraint qualifications in directionally differentiable multiobjective optimization problems." RAIRO - Operations Research 38.3 (2010): 255-274. <http://eudml.org/doc/105314>.

@article{Giorgi2010,
abstract = { 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. },
author = {Giorgi, Giorgio, Jiménez, Bienvenido, Novo, Vincente},
journal = {RAIRO - Operations Research},
keywords = {Multiobjective optimization problems; constraint qualification; necessary conditions for Pareto minimum; Lagrange multipliers; tangent cone; Dini differentiable functions; Hadamard differentiable functions; quasiconvex functions.; Multiobjective optimization problems, constraint qualification, necessary conditions for Pareto minimum, Lagrange multipliers, tangent cone, Dini differentiable functions, Hadamard differentiable functions, quasiconvex functions.},
language = {eng},
month = {3},
number = {3},
pages = {255-274},
publisher = {EDP Sciences},
title = {On constraint qualifications in directionally differentiable multiobjective optimization problems},
url = {http://eudml.org/doc/105314},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Giorgi, Giorgio
AU - Jiménez, Bienvenido
AU - Novo, Vincente
TI - On constraint qualifications in directionally differentiable multiobjective optimization problems
JO - RAIRO - Operations Research
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 3
SP - 255
EP - 274
AB - 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.
LA - eng
KW - Multiobjective optimization problems; constraint qualification; necessary conditions for Pareto minimum; Lagrange multipliers; tangent cone; Dini differentiable functions; Hadamard differentiable functions; quasiconvex functions.; Multiobjective optimization problems, constraint qualification, necessary conditions for Pareto minimum, Lagrange multipliers, tangent cone, Dini differentiable functions, Hadamard differentiable functions, quasiconvex functions.
UR - http://eudml.org/doc/105314
ER -

References

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  6. Y. Ishizuka, Optimality conditions for directionally differentiable multiobjective programming problems. J. Optim. Theory Appl.72 (1992) 91–111.  Zbl0793.90065
  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.  Zbl1010.90073
  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.  Zbl1097.90047
  10. T. Maeda, Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl.80 (1994) 483–500.  Zbl0797.90083
  11. O.L. Mangasarian, Nonlinear programming. McGraw-Hill, New York (1969).  Zbl0194.20201
  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 Operacional25 (2004) 34–47.  Zbl1097.90049
  13. V. Preda and I. Chitescu, On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl.100 (1999) 417–433.  Zbl0915.90231
  14. R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970).  Zbl0193.18401

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