The Notion of Invexity in Vector Optimization: Smooth and Nonsmooth Case

  • Giorgio Giorgi
  • Angelo Guerraggio
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 27)


After a survey of the main definitions of generalized convex and generalized invex vector functions, some other broad classes of generalized invex vector functions are introduced, both in the differentiable case and in the nonsmooth case. With reference to the said functions we extend some results of weak efficiency, efficiency and duality.


Vector Optimization Feasible Point Weak Duality Vector Minimum Invex Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ben-Israel A. and Mond B. (1986). What is invexity?. J. Austral. Math. Soc., Ser. B, 28, 1–9.MathSciNetMATHCrossRefGoogle Scholar
  2. Cambini A. and Martein L. (1994). Generalized concavity and optimality conditions in vector and scalar optimization; in Generalized Convexity (S. Komlósi, T. Rapcsàk and S. Schaible Eds.), Springer Verlag, Berlin, 337–357.Google Scholar
  3. Cambini R. (1996). Some new classes of generalized concave vector-valued functions. Optimization, 36, 11–24.MathSciNetMATHCrossRefGoogle Scholar
  4. Cambini R. and Komlósi S. (1996). Generalized concavity and generalized monotonicity concepts for vector valued functions. Report n.98, Dip. di Statistica e Matematica Applicata all’Economia, Università di Pisa.Google Scholar
  5. Clarke F.H. (1983). Optimization and nonsmooth analysis. J. Wiley; Sons, New York.MATHGoogle Scholar
  6. Craven B.D. (1978). Mathematical programming and control theory. Chapman and Hall, London.MATHGoogle Scholar
  7. Craven B.D. (1981). Vector-valued optimization; in Generalized Concavity in Optimization and Economics (S. Schaible and W.T. Ziemba Eds.), Academic Press, New York, 661–687.Google Scholar
  8. Craven B.D. and Glover B.M. (1985). Invex functions and duality. J. Austral. Math. Soc., Ser. A, 39, 1–20.MathSciNetMATHCrossRefGoogle Scholar
  9. Giorgi G. and Guerraggio A. (1996). Various types of nonsmooth invexity, Journal of Information and Optimization Sciences, 17, 137–150.MathSciNetMATHGoogle Scholar
  10. Jahn J. and Sacks E. (1986). Generalized quasi-convex mappings and vector optimization, Siam J. on Control and Optimization, 24, 306–322.MATHCrossRefGoogle Scholar
  11. Jeyakumar V. and Mond B. (1992). On generalized convex mathematical programmin,. J. Austral. Math. Soc., Ser. B, 34, 43–53.MathSciNetMATHCrossRefGoogle Scholar
  12. Lee G.M. (1994). Nonsmooth invexity in multiojective programming, Journal of Information and Optimization Sciences, 15, 127–136.MATHGoogle Scholar
  13. Martein L. (1994). Soluzioni efficienti e condizioni di ottimalità nell’ottimizzazione vettoriale; in Metodi di Ottimizzazione per le Decisioni (G. Di Pillo Ed.), Masson, Milano, 215–241.Google Scholar
  14. Reiland T.W. (1989). Generalized invexity for nonsmooth vector-valued mappings, Numer. Fund. Anal, and Optimization, 10, 1191–1202.MathSciNetCrossRefGoogle Scholar
  15. Reiland T.W. (1990). Nonsmooth invexity. Bull. Austral. Math. Soc, 42, 437–446.MathSciNetMATHCrossRefGoogle Scholar
  16. Weir T, Mond B. and Craven B.D. (1986). On duality for weakly minimized vector-valued optimization problems. Optimization, 17, 711–721.MathSciNetMATHCrossRefGoogle Scholar
  17. Yang X.Q., (1993). Generalized convex functions and vector variational inequalities. J. Optimization Theory and Applications, 79, 563–580.MATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers. Printed in the Netherlands 1998

Authors and Affiliations

  • Giorgio Giorgi
    • 1
  • Angelo Guerraggio
    • 2
  1. 1.University of PaviaPaviaItaly
  2. 2.University of VareseVareseItaly

Personalised recommendations