Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T17:58:25.110Z Has data issue: false hasContentIssue false
  • English
  • Français

On constraint qualifications in directionally differentiablemultiobjective optimization problems

Published online by Cambridge University Press:  15 September 2004

Giorgio Giorgi ,
Bienvenido Jiménez  and
Vincente Novo
Giorgio Giorgi
Affiliation:
Dipartimento di Ricerche Aziendali, Università degli Studi di Pavia, Via S. Felice, 5, 27100 Pavia, Italy; [email protected].
Bienvenido Jiménez
Affiliation:
Departamento de Economía e Historia Económica, Facultad de Economía y Empresa, Universidad de Salamanca, Campus Miguel de Unamuno, s/n, 37007 Salamanca, Spain; [email protected].
Vincente Novo
Affiliation:
Departamento de Matemática Aplicada, UNED, Calle Juan del Rosal, 12, Ciudad Universitaria, Apartado 60149, 28080 Madrid, Spain; [email protected].
Get access

Abstract

We consider a multiobjective optimization problem with a feasible setdefined by inequality and equality constraints such that all functionsare, at least, Dini differentiable (in some cases, Hadamard differentiableand sometimes, quasiconvex). Several constraint qualifications are givenin such a way that generalize both the qualifications introduced by Maedaand the classical ones, when the functions are differentiable. Therelationships between them are analyzed. Finally, we give severalKuhn-Tucker type necessary conditions for a point to be Pareto minimumunder the weaker constraint qualifications here proposed.

Keywords

Multiobjective optimization problemsconstraint qualificationnecessaryconditions for Pareto minimumLagrange multiplierstangent coneDinidifferentiable functionsHadamard differentiable functionsquasiconvexfunctions.
Type
Research Article
Information
RAIRO - Operations Research , Volume 38 , Issue 3 , July 2004 , pp. 255 - 274
Copyright
© EDP Sciences, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.P. Aubin and H. Frankowska, Set-valued analysis. Birkhaüser, Boston (1990).
M.S. Bazaraa and C.M. Shetty, Foundations of optimization. Springer-Verlag, Berlin (1976).
M.S. Bazaraa and C.M. Shetty, Nonlinear programming. John Wiley & Sons, New York (1979).
V.F. Demyanov and A.M. Rubinov, Constructive nonsmooth analysis. Verlag Peter Lang, Frankfurt am Main (1995).
Giorgi, G. and Komlósi, S., Dini derivatives in optimization. Part I. Riv. Mat. Sci. Econom. Social. Anno 15 (1992) 3–30.
Ishizuka, Y., Optimality conditions for directionally differentiable multiobjective programming problems. J. Optim. Theory Appl. 72 (1992) 91–111. CrossRef
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.
Jiménez, B. and Novo, V., Alternative theorems and necessary optimality conditions for directionally differentiable multiobjective programs. J. Convex Anal. 9 (2002) 97–116.
Jiménez, B. and Novo, V., Optimality conditions in directionally differentiable Pareto problems with a set constraint via tangent cones. Numer. Funct. Anal. Optim. 24 (2003) 557–574. CrossRef
Maeda, T., Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl. 80 (1994) 483–500. CrossRef
O.L. Mangasarian, Nonlinear programming. McGraw-Hill, New York (1969).
Novo, V. and Jiménez, B., 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.
Preda, V. and Chitescu, I., On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl. 100 (1999) 417–433. CrossRef
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970).