Journal of Global Optimization

, Volume 52, Issue 3, pp 391–409 | Cite as

Convex envelopes of products of convex and component-wise concave functions

  • Aida Khajavirad
  • Nikolaos V. Sahinidis


In this paper, we consider functions of the form \({\phi(x,y)=f(x)g(y)}\) over a box, where \({f(x), x\in {\mathbb R}}\) is a nonnegative monotone convex function with a power or an exponential form, and \({g(y), y\in {\mathbb R}^n}\) is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.


Convex envelope Global optimization Factorable programming Submodular functions 


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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA

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