Global optimization of polynomials using generalized critical values and sums of squares

  • Authors:

  • Feng Guo;Mohab Safey El Din;Lihong Zhi

  • Affiliations:

  • Key Laboratory of Mathematics Mechanization, AMSS, Beijing, China;UPMC, Univ Paris, INRIA, Paris-Rocquencourt Center, France;Key Laboratory of Mathematics Mechanization, AMSS, Beijing, China

  • Venue:
  • Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
  • Year:
  • 2010

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Abstract

Let &Xmacr; = [X1, ..., Xn] and f ∈ R[&Xmacr;]. We consider the problem of computing the global infimum of f when f is bounded below. For A ∈ GLn(C), we denote by fA the polynomial f(A &Xmacr;). Fix a number M ∈ R greater than infx∈Rn f(x). We prove that there exists a Zariski-closed subset A [equation] GLn(C) such that for all A ∈ GLn(Q) \ A, we have fA ≥ 0 on Rn if and only if for all ε 0, there exist sums of squares of polynomials s and t in R[&Xmacr;] and polynomials [Equation]. Hence we can formulate the original optimization problems as semidefinite programs which can be solved efficiently in Matlab. Some numerical experiments are given. We also discuss how to exploit the sparsity of SDP problems to overcome the ill-conditionedness of SDP problems when the infimum is not attained.