What is invexity? (English) Zbl 0603.90119

Ben-Israel, A.; Mond, B.

J. Aust. Math. Soc., Ser. B 28, 1-9 (1986).

A differentiable function \(f: R^ n\to R\) is invex if there exists a vector function \(\eta\) (x,u) such that f(x)-f(u)\(\geq [\eta (x,u)]^ t\nabla f(u)\). It is shown that f is invex if and only if every stationary point is a global minimum. In particular, if f has no stationary points then f is invex.
A characterization of invex function is given and relationships of invex functions to other generalized-convex functions are discussed. Application to constrained optimization is discussed.
It is also shown that the Slater condition which is associated with convex functions also applies more generally in the same way when invex functions rather than convex functions are involved.

Reviewer: M.Hanson

MSC:

90C30	Nonlinear programming
49M37	Numerical methods based on nonlinear programming

Keywords:

invexity; generalized convexity; stationary point; global minimum; characterization of invex function; constrained optimization; Slater condition