Journal of Optimization Theory and Applications

, Volume 47, Issue 1, pp 65–76

The essence of invexity

  • D. H. Martin
Contributed Papers

DOI: 10.1007/BF00941316

Cite this article as:
Martin, D.H. J Optim Theory Appl (1985) 47: 65. doi:10.1007/BF00941316


The notion of invexity was introduced into optimization theory by Hanson in 1981 as a very broad generalization of convexity. A smooth mathematical program of the form minimizef(x), subject tog(x) ≦ 0, isxD ⊑ ℝninvex if there exists a function η:D ×D → ℝn such that, for allx, uD,
$$\begin{gathered} f(x) - f(u) - f'(u)n(x,u) \geqq 0, \hfill \\ g(x) - g(u) - g'(u)n(x,u) \geqq 0. \hfill \\ \end{gathered}$$

The convex case corresponds of course to η(x, u)≡xu; but, as Hanson showed, invexity is sufficient to imply both weak duality and that the Kuhn-Tucker conditions are sufficient for global optimality.

It is shown here that elementary relaxations of the conditions defining invexity lead to modified invexity notions which are both necessary and sufficient for weak duality and Kuhn-Tucker sufficiency.

Key Words

Invexityconvex mathematial programsKuhn-Tucker conditionsweak duality

Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • D. H. Martin
    • 1
  1. 1.National Research Institute for Mathematical Sciences of the CSIRPretoriaRepublic of South Africa