Original Paper

Journal of Global Optimization

, Volume 40, Issue 4, pp 697-718

First online:

Global minimization of rational functions and the nearest GCDs

  • Jiawang NieAffiliated withDepartment of Mathematics, University of California Email author 
  • , James DemmelAffiliated withDepartment of Math & EECS, University of California
  • , Ming GuAffiliated withDepartment of Mathematics, University of California

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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 function Polynomial Global minimization Sum of squares (SOS) Greatest common divisor Quadratic module