Pseudolinear function: Difference between revisions

Revision as of 20:21, 23 December 2010

Revision as of 20:32, 23 December 2010

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In mathematics, a '''pseudoconvex function''' <math>f:X\rightarrow\mathbb{R}</math> on aan open convex set <math>X\subseteq\mathbb{R}</math> is a function that is differentiable in <math>X</math> such that for every <math>x,y\in X</math>,

:<math>f\left(y\right)<f\left(x\right)\Rightarrow\left(y-x\right)^{T}\nabla f\left(x\right)<0. \, </math>

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A '''pseudolinear function''' is one that is both pseudoconvex and pseudoconcave.

It can be shown (see Cambini and Martein) that <math>f</math> is pseudolinear if and only if for every <math>x,y\in X</math>,

: <math> f(x) = f(y)\text{ if and only if }\nabla f(x)^T (y - x) = 0. \, </math>

== References ==