An Introduction to Abstract Convexity

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


One of the main results of convex analysis states that an arbitrary lower semicontinuous convex function f (perhaps admitting the value +∞) is the upper envelope of the set of all its affine minorants:
$$f\left( x \right) = \sup \left\{ {h\left( x \right):h\;is\;an\;affine\;function,\;h \leqslant f} \right\}$$


Convex Function Coupling Function Separation Property Affine Function Vertical Shift 
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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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