Elements of Monotonic Analysis: IPH Functions and Normal Sets

  • Alexander Rubinov
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 44)


In this and next chapters we examine some classes of monotonic functions defined on either the cone \(\mathbb{R}_{ + }^{n} = \left\{ {x \in {{\mathbb{R}}^{n}}:{{x}_{i}}0\;for\;all\;i = 1, \ldots ,n} \right\}\) or the cone \(\mathbb{R}_{ + + }^n = \left\{ {x \in \mathbb{R}_ + ^n:\;{x_i} > 0\;for\;all\;i = 1, \ldots ,n} \right\} \) and the so-called normal subsets of these cones. We shall consider these classes in the framework of abstract convexity. The simplest and most elegant theory can be obtained for functions defined on the cone \(\mathbb{R}_{ + + }^n \) and subsets of this cone. A nonempty set \(U \in \mathbb{R}_ + ^n \) is said to be normal if
$$ (x \in U, x' \in \mathbb{R}_ + ^n, x' \leqslant x) \Rightarrow x' \in U. $$


Abstract Convexity Sublinear Function Normal Subset Monotonic Analysis Normal Hull 
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Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Alexander Rubinov
    • 1
  1. 1.School of Information Technology and Mathematical SciencesUniversity of BallaratVictoriaAustralia

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