Journal of Global Optimization

, Volume 71, Issue 3, pp 571–592 | Cite as

Reformulations for utilizing separability when solving convex MINLP problems

  • Jan Kronqvist
  • Andreas Lundell
  • Tapio Westerlund


Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.


Convex MINLP Lifted polyhedral approximation Separable MINLP Extended cutting plane algorithm Extended supporting hyperplane algorithm Outer approximation 



Financial support from the Finnish Graduate School in Chemical Engineering is gratefully acknowledged, as is support from GAMS Development Corporation.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Jan Kronqvist
    • 1
  • Andreas Lundell
    • 2
  • Tapio Westerlund
    • 1
  1. 1.Process Design and Systems EngineeringÅbo Akademi UniversityTurkuFinland
  2. 2.Mathematics and StatisticsÅbo Akademi UniversityTurkuFinland

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