IJPAM: Volume 10, No. 1 (2004)

MINIMUM PRINCIPLE-TYPE OPTIMALITY
CONDITIONS FOR PARETO PROBLEMS

G. Giorgi$^1$, B. Jiménez$^2$, V. Novo$^3$
$^1$Dipartimento di Ricerche Aziendali
Università Degli Studi di Pavia
Via S. Felice 5, 27100 Pavia, ITALY
e-mail: ggiorgi@eco.unipv.it
$^2$ Departamento de Economía e Historia Económica
Facultad de Economía y Empresa
Universidad de Salamanca
Campus Miguel de Unamuno
s/n, 37007, Salamanca, SPAIN
e-mail: bjimen1@encina.pntic.mec.es
$^3$Departamento de Matemática Aplicada
Escuela Técnica Superior de Ingenieros Industriales
Universidad Nacional de Educación a Distancia
C/ Juan del Rosal, 12 Ciudad Universitaria
Apartado de correos 60149, 28080 Madrid, SPAIN
e-mail: vnovo@ind.uned.es


Abstract.We study necessary optimality conditions for Pareto problems with three kinds of constraints: inequality constraints, equality constraints and a set constraint. We suppose that the objective function and the inequality constraints are Hadamard (directionally) differentiable at the optimal solution and the equality constraints are continuous around and Fréchet differentiable at the optimal solution. We provide minimum principle necessary optimality conditions for a problem with a convex set constraint whose interior may be empty. Some constraint qualifications are considered to get Kuhn-Tucker conditions. We also provide minimum principle necessary optimality conditions for a problem with an arbitrary set constraint under a generalized Bender constraint qualification (GBCQ), which is automatically satisfied by the interior tangent cone and the cone of quasi-interior directions to the constraint set.

Received: October 9, 2003

AMS Subject Classification: 90C29, 90C46

Key Words and Phrases: Pareto problems, minimum principle, Fritz John and Kuhn-Tucker optimality conditions, tangent cone, generalized differentiability

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 10
Issue: 1