Higher order invex functions and higher order pseudoinvex ones

Abstract

In this article we introduce a notion of a higher order invex scalar function in terms of the higher order lower Dini directional derivatives. This notion differs from the respective ones, applied in duality theory. Higher order invex functions are expanding classes of functions in the sense that every invex function of order n (n is a positive integer) is invex of order (n + 1). We prove that a function f is invex of order n if and only if the set of stationary points of order n of f coincides with the set of global minimizers. We extend the known property that, if we do not specify the kernel η, then a differentiable function is invex if and only if it is pseudoinvex, to a result, which includes higher order Dini derivatives. We introduce a notion of a pseudoinvex function of order n with respect to a known map η. The pseudoinvex functions of order n are also expanding classes, intermediate between pseudoinvex and prequasiinvex functions. Further, we obtain characterizations of the solution set of the minimization problem of a pseudoinvex function of order n over an invex set, provided that a fixed solution is known. Some known results become particular cases of our theorems.

Keywords:

• nonsmooth analysis
• nonsmooth optimization
• invex functions
• characterizations of the solution set of the nonlinear programming problem

AMS Subject Classifications::

• 26B25
• 49J52
• 90C26

Acknowledgements

This research is partially supported by the TU Varna Grant No 18/2012.