Strict and stable generalization of convex functions and monotone maps

A function f is said to be stable with respect to some property (P) if there exists epsilon > 0 such that f + xi fulfill (P) for all linear functional xi satisfying parallel xi parallel < epsilon. S-quasiconvex functions introduced by Phu (Optimization, Vol.38, 1996) are stable with respect to the properties: 'every lower level set is convex', 'each local minimizer is a global minimizer', and 'each stationary point is a global minimizer'. Correspondingly, we introduced the concepts of s-quasimonotone maps and showed that in the case of a differentiable map, s-quasimonotonicity of the gradient is equivalent to s-quasiconvexity of the underlying function. In this paper, strictly s-quasiconvex functions and strictly s-quasimonotone maps are introduced. In the case of a differentiable map, strict s-quasimonotonicity of the gradient is equivalent to strict s-quasiconvexity of the underlying function, too. An algorithm for finding supremum of the set of all epsilon above of a continuously twice differentiable strictly s-quasiconvex function on R sup 1 is presented.