Taiwanese Journal of Mathematics


B. D. Craven

Full-text: Open access


Necessary Lagrangian conditions for a constrained minimum become sufficient under generalized convex assumptions, in particular invex, and duality results follow. Many classes of vector functions with properties related to invex have been studied, but It has not been clear how far these classes are distinct. Various inclusions between these classes are now established Some modifications of invex can be regarded as perturbations of invex. There is a stability criterion for when the invex property is preserved under small perturbations. Some results extend to nondifferentiable (Lipschitz) functions.

Article information

Taiwanese J. Math., Volume 14, Number 5 (2010), 1925-1933.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Craven, B. D. KINDS OF VECTOR INVEX. Taiwanese J. Math. 14 (2010), no. 5, 1925--1933. doi:10.11650/twjm/1500406024. https://projecteuclid.org/euclid.twjm/1500406024

Export citation


  • [1.] E. Caprari, $\rho$-invex functions and $({\cal F},\rho)-$ convex functions: properties and equivalences, Optimization, 52(1) (2003), 65-74.
  • [2.] Xiuhong Chen, Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl., 290(2) (2004), 423-435.
  • [3.] B. D. Craven, Relations between invex propertiex, WSSIAA, 5 (1995), 25-34, World Scintific Publishing Company, Singapore.
  • [4.] B. D. Craven, Non-differentiable invex, Proceedings of the Centre for Mathematics and ite Applications, Australian National University, Canberra, Australia, 36 (1999), 25-28.
  • [5.] B. D. Craven, Vector generalized invex, Opsearch, 38(4) (2001), 345-361.
  • [6.] B. D. Craven, Global invexity and duality in mathematical programming, Asia-Pacific Journal of Operational Research, 19 (2002), 169-175.
  • [10.] V. Jeyakumar and B. Mond, On generalized convex mathematical progamming, J. Austral. Math. Soc., Ser. B, 34 (1992), 43-53. \item [11.] B. Mond and T. Weir, Generalized convexity and duality, in: Generalized Convexity in Optimization and Economics, S. Schaible and W. T. Ziemba (eds), Academic Press, New York, 1981, pp. 263-280. \item [12.] S. M. Robinson, Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems, SIAM J. Numerical Analysis, 13 (1976), 497-513. \item [13.] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl., 126 (1988), 29-38. \item [14.] C., Z\vaa linescu, A critical view on invexity, Journal, of,Optimization, Theory, and Applications, to appear.