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Journal of Global Optimization

, Volume 32, Issue 3, pp 367–383 | Cite as

Multiplier Rules and Saddle-Point Theorems for Helbig’s Approximate Solutions in Convex Pareto Problems

  • César GutiérrezEmail author
  • Bienvenido Jiménez
  • Vicente Novo
Article

Abstract

This paper deals with approximate Pareto solutions in convex multiobjective optimization problems. We relate two approximate Pareto efficiency concepts: one is already classic and the other is due to Helbig. We obtain Fritz John and Kuhn–Tucker type necessary and sufficient conditions for Helbig’s approximate solutions. An application we deduce saddle-point theorems corresponding to these solutions for two vector-valued Lagrangian functions.

Keywords

Approximate solutions Lagrangian functions ε-Pareto optimality ε-Saddle-points ε-Subdifferential 

Mathematics Subject Classifications

90C29 49M37 

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References

  1. Corley, H.W. 1981Duality theory for maximization with respect to conesJournal of Mathematical Analysis and Applications84560568CrossRefGoogle Scholar
  2. Helbig, S. (1992), On a new concept for ε-efficiency, In: Optimization Days 1992, Montreal, Canada.Google Scholar
  3. Gutiérrez, C., Jiménez, B and Novo, V., ε-Pareto optimality conditions for convex multiobjective programming via max function, submitted paper.Google Scholar
  4. Hiriart-Urruty, J.-B. 1982ε-Subdifferential calculusAubin, J.-P.Vinter, R.B. eds. Convex Analysis and Optimization. Research Notes in Mathematical Series No. 57Pitman PublishersNew York4392Google Scholar
  5. Kutateladze, S.S. 1979Convex ε-programmingSoviet Math. Dokl.20391393Google Scholar
  6. Liu, J.C. 1991ε-Duality theorem of nondifferentiable nonconvex multiobjective programmingJournal of Optimization Theory and Applications69153167CrossRefGoogle Scholar
  7. Liu, C. 1996ε-Pareto optimality for nondifferentiable multiobjective programming via penalty functionJournal of Mathematical Analysis and Applications198248261CrossRefGoogle Scholar
  8. Liu, J.C., Yokoyama, K. 1999ε-Optimality and duality for multiobjective fractional programmingComputers and Mathematics with Applications37119128CrossRefGoogle Scholar
  9. Loridan, P. 1982Necessary conditions for ε-optimalityMathematical Programming Study19140152Google Scholar
  10. Loridan, P. 1984ε-Solutions in vector minimization problemsJournal of Optimization Theory and Applications43265276CrossRefGoogle Scholar
  11. Luc, D.T. 1984On duality theory in multiobjective programmingJournal of Optimization Theory and Applications43557582CrossRefGoogle Scholar
  12. Luenberger, D.G. 1969Optimization by Vector Space MethodsJohn Wiley & SonsNew YorkGoogle Scholar
  13. Rong, W.D., Wu, Y.N. 2000ε-Weak minimal solutions of vector optimization problems with set-valued mapsJournal of Optimization Theory and Applications106569579CrossRefGoogle Scholar
  14. Strodiot, J.J., Nguyen, V.H., Heukemes, N. 1983ε-Optimal solutions in nondifferentiable convex programming and some related questionsMathematical Programming25307328Google Scholar
  15. Tanino, T., Sawaragi, Y. 1979Duality theory in multiobjective programmingJournal of Optimization Theory and Applications27509529CrossRefGoogle Scholar
  16. Vályi, I. 1987Approximate saddle-point theorems in vector optimizationJournal of Optimization Theory and Applications55435448CrossRefGoogle Scholar
  17. Yokoyama, K. 1992ε-Optimality criteria for convex programming problems via exact penalty functionsMathematical Programming56233243CrossRefGoogle Scholar
  18. Yokoyama, K. 1996Epsilon approximate solutions for multiobjective programming problemsJournal of Mathematical Analysis and Applications203142149CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • César Gutiérrez
    • 1
    Email author
  • Bienvenido Jiménez
    • 2
  • Vicente Novo
    • 3
  1. 1.Departamento de Matemática Aplicada, ETSII, Edificio de Tecnologías de la Información y las TelecomunicacionesUniversidad de ValladolidValladolidSpain
  2. 2.Departamento de Economía e Historia Económica, Facultad de Economía y EmpresaUniversidad de SalamancaSalamancaSpain
  3. 3.Departamento de Matemática Aplicada, ETSIIUniversidad Nacional de Educación a DistanciaMadridSpain

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