Journal of Global Optimization

, Volume 60, Issue 2, pp 217–238 | Cite as

Extended formulations for convex envelopes

  • Martin Ballerstein
  • Dennis Michaels


In this work we derive explicit descriptions for the convex envelope of nonlinear functions that are component-wise concave on a subset of the variables and convex on the other variables. These functions account for more than 30 % of all nonlinearities in common benchmark libraries. To overcome the combinatorial difficulties in deriving the convex envelope description given by the component-wise concave part of the functions, we consider an extended formulation of the convex envelope based on the Reformulation–Linearization-Technique introduced by Sherali and Adams (SIAM J Discret Math 3(3):411–430, 1990). Computational results are reported showing that the extended formulation strategy is a useful tool in global optimization.


Convex envelope Edge-concave functions Extended formulation Reformulation–Linearization-Technique Simultaneous convexification 



This work is part of the Collaborative Research Centre “Integrated Chemical Processes in Liquid Multiphase Systems” (CRC/Transregio 63 “InPROMPT”) funded by the German Research Foundation (DFG). Main parts of this work have been finished while the second author was at the Institute for Operations Research at ETH Zurich and financially supported by DFG through the CRC/Transregio 63. The authors thank the DFG for its financial support. We would like to thank Jon Lee for providing the Lavor test instances used in this paper. We are grateful to Stefan Vigerske for his continued support for SCIP.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Eidgenössische Technische Hochschule ZürichInstitut für Operations ResearchZurichSwitzerland
  2. 2.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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