Journal of Optimization Theory and Applications

, Volume 47, Issue 1, pp 65–76 | Cite as

The essence of invexity

  • D. H. Martin
Contributed Papers

Abstract

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

Invexity convex mathematial programs Kuhn-Tucker conditions weak duality 

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References

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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

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