Mathematical Programming

, Volume 93, Issue 2, pp 247–263 | Cite as

Convex extensions and envelopes of lower semi-continuous functions

  • Mohit Tawarmalani
  • Nikolaos V Sahinidis


 We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms.


Convex Function Nonlinear Function Extension Theory Convex Relaxation Convex Envelope 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Mohit Tawarmalani
    • 1
  • Nikolaos V Sahinidis
    • 2
  1. 1.Krannert School of Management, Purdue University. e-mail:
  2. 2.Department of Chemical Engineering, University of Illinois at Urbana-Champaign. e-mail:

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