Mathematical Programming

, Volume 169, Issue 1, pp 69–94 | Cite as

DC decomposition of nonconvex polynomials with algebraic techniques

  • Amir Ali Ahmadi
  • Georgina Hall
Full Length Paper Series B


We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) and find ones that speed up the convex–concave procedure. We prove, however, that optimizing over the entire set of dcds is NP-hard.


Difference of convex programming Conic relaxations Polynomial optimization Algebraic decomposition of polynomials 

Mathematics Subject Classification

90C22 90C26 90C60 68Q25 



We would like to thank Pablo Parrilo for insightful discussions and Mirjam Dür for pointing out reference [9].


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.ORFEPrinceton UniversityPrincetonUSA

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