Journal of Global Optimization

, Volume 40, Issue 4, pp 697–718

Global minimization of rational functions and the nearest GCDs


    • Department of MathematicsUniversity of California
  • James Demmel
    • Department of Math & EECSUniversity of California
  • Ming Gu
    • Department of MathematicsUniversity of California
Original Paper

DOI: 10.1007/s10898-006-9119-8

Cite this article as:
Nie, J., Demmel, J. & Gu, M. J Glob Optim (2008) 40: 697. doi:10.1007/s10898-006-9119-8


This paper discusses the global minimization of rational functions with or without constraints. We propose sum of squares relaxations to solve these problems, and study their properties. Some special features are discussed. First, we consider minimization of rational functions without constraints. Second, as an application, we show how to find the nearest common divisors of polynomials via unconstrained minimization of rational functions. Third, we discuss minimizing rational functions under some constraints which are described by polynomials.


Rational functionPolynomialGlobal minimizationSum of squares (SOS)Greatest common divisorQuadratic module

Copyright information

© Springer Science+Business Media, Inc. 2006