函数强伪凸性与映射强伪单调性

1　引　　言Schaible在[1]中引入七类单调映射,并证明对其中六类,函数的某种广义凸性都和相应的梯度单调性等价,只有函数强伪凸和梯度强伪单调的等价性是否成立作为公开问题.其后,Schaible又在[2]中通过一个例子否定了两者的等价性,并引入了较弱的函数强伪凸和映射强伪单调的概念,在函数二次可微的条件下证明了函数强伪凸和梯度强伪单调等价.我们将引入强于[2]中概念的强伪凸和强伪单调性,对给出的定义,不附加条件便可保证函数强伪凸性和梯度强伪单调性等价.同时,对[2]中的一个错误予以指出,并给出正确的反例.还就[1]中问题给出远比[2]中简…

ON STRONG PSEUDOCONVEXITY AND STRONG PSEUDOMONOTONICITY

New concepts of strongly pseudoconvex functions and strongly pseudomonotone maps are introduced, which are stronger than corresponding concepts in ref.2. It is shown that strong pseudomonotonicity of the gradient corresponds to strong pseudoconvexity of the underlying function, without any other assumption of the function. Point out the error example 3.1 in ref.2, and give a right counterexample. Also, for an open question in ref.1, a very simple counterexample is presented.