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Monotonicity in the Framework of Generalized Convexity

  • Hoang Tuy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 77)

Abstract

An increasing function f : R n R ∪ +∞ is a function such that f(x′) f(x) whenever x′ x (component-wise). A downward set G R n is a set such that x G whenever x′ x for some x′ G. We present a geometric theory of monotonicity in which increasing functions relate to downward sets in the same way as convex functions relate to convex sets. By giving a central role to a separation property of downward sets similar to that of convex sets, a theory of monotonic optimization can be developed which parallels d.c. optimization in several respects.

Keywords

Monotonicity Downward sets Normal sets Separation property Polyblock Increasing functions Monotonic functions Difference of monotonic functions (d.m. functions) Abstract convex analysis Global optimization 

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsVietnam

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